Testing the Combination of the Anticancer Drugs Trastuzumab Deruxtecan (DS-8201a) and Azenosertib (ZN-c3) in Patients With Stomach or Other Solid Tumors

Inclusion Criteria: * In the dose escalation, patients must have a histologically documented locally advanced, unresectable, or metastatic solid tumor that has progressed following at least one prior line of treatment in the metastatic setting or has no satisfactory alternative treatment option and all of the following: * HER2 expression by immunohistochemistry (IHC) (1+, 2+, or 3+) or HER2 amplification by in situ hybridization (ISH) or next generation sequencing (NGS) (on any Clinical Laboratory Improvements Amendments \[CLIA\] platform), AND * T-DXd (DS-8201a)-naive disease * In the dose expansion, patients must have histologically documented locally advanced, unresectable or metastatic gastric or gastroesophageal junction (GEJ) cancer that has progressed following at least one prior line of treatment in the metastatic setting and have all of the following: * CCNE1 amplification, AND * HER2 expression by IHC (1+, 2+, or 3+) or HER2 amplification by ISH or NGS (on any CLIA platform), AND * T-DXd (DS-8201a)-naive disease * Received prior trastuzumab-based treatment, if eligible for such treatment * For the dose escalation and dose expansion, patients can have evaluable or measurable disease * Potential trial participants should have recovered from clinically significant adverse events (AEs) of their most recent therapy/intervention prior to enrollment * Age ≥ 18 years. Because no dosing or AE data are currently available on the use of T-DXd (DS-8201a) in combination with azenosertib (ZN-c3) in patients \< 18 years of age, children are excluded from this study * Eastern Cooperative Oncology Group (ECOG) performance status of ≤ 1 (Karnofsky ≥ 70%). Both T-DXd (DS-8201a) and azenosertib (ZN-c3) have fatigue as an adverse effect. Due to the overlapping adverse effect, the performance status cannot be less restrictive * Absolute neutrophil count ≥ 1.5 × 10\^9/L (within 7 days of study treatment initiation) * No transfusions with red blood cells or platelets are allowed within 1 week prior to screening assessment * No administration of granulocyte colony-stimulating factor is allowed within 1 week prior to screening assessment * Hemoglobin \> 9.0 g/dL (within 7 days of study treatment initiation) * No transfusions with red blood cells or platelets are allowed within 1 week prior to screening assessment * No administration of granulocyte colony-stimulating factor is allowed within 1 week prior to screening assessment * Platelets ≥ 100 × 10\^9/L (within 7 days of study treatment initiation) * No transfusions with red blood cells or platelets are allowed within 1 week prior to screening assessment * No administration of granulocyte colony-stimulating factor is allowed within 1 week prior to screening assessment * Total bilirubin ≤ 1.5 institutional upper limit of normal (ULN). Documented Gilbert syndrome is allowed if total bilirubin is ≤ 3 × institutional ULN (within 7 days of study treatment initiation) * No transfusions with red blood cells or platelets are allowed within 1 week prior to screening assessment * No administration of granulocyte colony-stimulating factor is allowed within 1 week prior to screening assessment * Aspartate aminotransferase (AST \[serum glutamic-oxaloacetic transaminase (SGOT)\])/alanine aminotransferase (ALT \[serum glutamate pyruvate transaminase (SGPT)\]) ≤ 3 × institutional ULN. In the presence of liver metastases, AST or ALT up to 5 × institutional ULN is permitted (within 7 days of study treatment initiation) * No transfusions with red blood cells or platelets are allowed within 1 week prior to screening assessment * No administration of granulocyte colony-stimulating factor is allowed within 1 week prior to screening assessment * Measured of calculated creatinine clearance (CrCl) ≥ 60 mL/min (CrCl should be calculated per institutional standard; glomerular filtration rate can also be used in place of CrCl) ≥ 60 mL/min for patients with creatinine levels \> 1.5 x institutional ULN (within 7 days of study treatment initiation) * No transfusions with red blood cells or platelets are allowed within 1 week prior to screening assessment * No administration of granulocyte colony-stimulating factor is allowed within 1 week prior to screening assessment * International normalized ratio/prothrombin time and activated partial thromboplastin time ≤ 1.5 × institutional ULN (within 7 days of study treatment initiation) * No transfusions with red blood cells or platelets are allowed within 1 week prior to screening assessment * No administration of granulocyte colony-stimulating factor is allowed within 1 week prior to screening assessment * Patients must have left ventricular ejection fraction (LVEF) ≥ 50% by either an echocardiogram (ECHO) or multigated acquisition (MUGA) scan within 28 days before enrollment * Human immunodeficiency virus-infected patients on effective anti-retroviral therapy with undetectable viral load within 6 months are eligible for this trial * For patients with evidence of chronic hepatitis B virus (HBV) infection, the HBV viral load must be undetectable on suppressive therapy, if indicated * Patients with a history of hepatitis C virus (HCV) infection must have been treated and cured. For patients with HCV infection who are currently on treatment, they are eligible if they have an undetectable HCV viral load * Patients with treated brain metastases are eligible if follow-up brain imaging after central nervous system (CNS)-directed therapy shows no evidence of progression * Patients with new or progressive brain metastases (active brain metastases) or leptomeningeal disease are eligible if the treating physician determines that immediate CNS-specific treatment is not required and is unlikely to be required during the first cycle of study treatment * Patients with a prior or concurrent malignancy whose natural history or treatment does not have the potential to interfere with the safety or efficacy assessment of the investigational regimen are eligible for this trial * Life expectancy ≥ 3 months * Women of childbearing potential (WOCBP) must have a negative serum pregnancy test result within 3 days of study treatment initiation * Agents composed of HER2 antibody conjugated to a topoisomerase 1 inhibitor and azenosertib (ZN-c3) are known to be teratogenic; thus, WOCBP must agree to use highly effective contraception from time of screening and throughout the study treatment period and for at least 7 months after final study treatment administration. Should a woman become pregnant or suspect she is pregnant while she or her partner is participating in this study, she should inform her treating physician immediately. Female patients must not donate, or retrieve for their own use, ova from the time of screening and throughout the study treatment period and for at least 7 months after the final study treatment administration * Women of non-childbearing potential defined as premenopausal females with documented tubal ligation or hysterectomy; or postmenopausal defined as 12 months of spontaneous amenorrhea (in questionable cases, a blood sample with simultaneous follicle-stimulating hormone \> 40 mIU/mL and estradiol \< 40 pg/mL \[\< 147 pmol/L\] is confirmatory) are eligible. Females on hormone replacement therapy (HRT) and whose menopausal status is in doubt will be required to use one of the contraception methods outlined for WOCBP if they wish to continue their HRT during the study. Otherwise, they must discontinue HRT to allow confirmation of postmenopausal status prior to study enrollment. For most forms of HRT, at least 2-4 weeks will elapse between the cessation of therapy and the blood draw; this interval depends on the type and dosage of HRT. Following confirmation of their postmenopausal status, they can resume use of HRT during the study without use of a contraception method * Male patients involved with WOCBP must agree to use a highly effective form of contraception or avoid intercourse from time of screening and throughout the study treatment period and for at least 4 months after the last dose of study treatment. Male patients must not freeze or donate sperm starting at screening and throughout the study period and at least 4 months after the final study treatment administration. Preservation of sperm should be considered prior to enrollment in this study * Ability to understand and the willingness to sign a written informed consent document. Legally authorized representatives may sign and give informed consent on behalf of study participants * Willing to undergo biopsy as required by the study (dose expansion only) Exclusion Criteria: * As azenosertib (ZN-c3) is a substrate of CYP3A4, use of prescription or non-prescription drugs known to be moderate or strong inhibitors or inducers of CYP3A4 are prohibited with the exception of moderate or strong inhibitors or inducers of CYP3A4 that are part of the prophylactic antiemetic regimen. Chloroquine/hydroxychloroquine are metabolized by CYP3A4 and therefore, should be prohibited. For patients who have received prior moderate or strong inhibitors or inducers of CYP3A4, the required washout period is approximately 5 half-lives prior to study treatment initiation * Patients who are receiving any other investigational agents * Patients who have a history of allergic reactions attributed to compounds of similar chemical or biologic composition to the study drugs * Patients who have a history of severe hypersensitivity reactions to other monoclonal antibodies (mAbs) * Patients with clinically severe pulmonary compromise resulting from intercurrent pulmonary illnesses including, but not limited to, any underlying pulmonary disorder (i.e., pulmonary emboli within 3 months of the study enrollment, severe asthma, severe chronic obstructive pulmonary disease, restrictive lung disease, pleural effusion, etc.), and any autoimmune, connective tissue or inflammatory disorders with potential pulmonary involvement (i.e., rheumatoid arthritis, Sjogren's, sarcoidosis, etc.), or prior pneumonectomy * Pregnant women are excluded from this study because T-DXd (DS-8201a) and azenosertib (ZN-c3) have the potential risk for teratogenic or abortifacient effects. Because there is an unknown but potential risk for AEs in nursing infants secondary to treatment of the mother with T-DXd (DS-8201a) and azenosertib (ZN-c3), breastfeeding should be discontinued if the mother is treated with T-DXd (DS-8201a) or azenosertib (ZN-c3) * Patients with history of non-infectious pneumonitis/interstitial lung disease (ILD), current ILD, or where suspected ILD cannot be ruled out by imaging at screening * Patients with active infections requiring antibiotics at the time of study treatment initiation are not eligible * Patients with history of malabsorption syndrome or other condition that would interfere with enteral absorption or results in the inability or unwillingness to swallow pills * Patients with current signs or symptoms of bowel obstruction including sub-occlusive disease related to underlying disease * Patients with a medical history of myocardial infarction within 6 months before enrollment, symptomatic congestive heart failure (New York Heart Association class IIb to IV), and/or troponin levels consistent with myocardial infarction as defined according to the manufacturer 28 days prior to enrollment * Patients with clinically significant corneal disease * Patients with a pleural effusion, ascites, or pericardial effusion that requires drainage, peritoneal shunt, or cell-free and concentrated ascites reinfusion therapy (CART). (Drainage and CART are not allowed within 2 weeks prior to screening assessment) * Patients with history of Torsades de Pointes unless all risk factors that contributed to Torsades de Pointes have been corrected * Based on an average of triplicate 12-lead electrocardiogram (ECG), patients with a mean resting corrected QT (QTc) interval using Fridericia formula of \> 470 msec for both males and females at screening or a history of congenital long QT syndrome will be excluded * Patients with prior treatment with a WEE1 inhibitor (dose escalation and dose expansion) * Patients with prior treatment with T-DXd (DS-8201a) or other topoisomerase inhibitors (dose escalation and dose expansion) * Patients with uncontrolled intercurrent illness * Patients with prior allogeneic organ transplantation including allogeneic stem cell transplantation * Patients with clinically significant chronic gastrointestinal disorder with diarrhea as a major symptom; ≥ grade 2 diarrhea at baseline. Please contact the protocol principal investigator (PI) for any patient with more than two episodes of diarrhea per day averaged over at least a 7-day period at time of screening to determine whether the diarrhea would be considered clinically significant * Patients with spinal cord compression