Testing Different Amounts of the Combination of Drugs M1774 and ZEN-3694 for the Treatment of Recurrent Ovarian and Endometrial Cancer

Inclusion Criteria: * Patients must have pathologically confirmed: * PART I: Recurrent clear cell or endometrioid ovarian carcinoma (at least 50% morphology of clear cell and endometrioid required), recurrent clear cell and low grade endometrioid endometrial carcinoma (The International Federation of Gynecology and Obstetrics \[FIGO\] grade 1), or recurrent platinum resistant high grade serous ovarian carcinoma * NOTE: platinum-resistant disease is defined as progression within \< 6 months from completion of platinum-based therapy. The date should be calculated from the last administered dose of platinum therapy * NOTE: Institutional pathology reports must be provided indicating at least 50% endometrioid or clear cell morphology for ovarian cancer. * NOTE: Patients with recurrent endometrial carcinoma must not be eligible for or decline treatment with curative intent. * PART II: Recurrent clear cell or endometrioid ovarian carcinoma (at least 50% tumor morphology of clear cell and endometrioid required). Recurrent clear cell or FIGO Grade 1 endometrioid endometrial carcinoma. Next Generation Sequencing (NGS) by Clinical Laboratory Improvement Act (CLIA) approved lab required for ARID1A status. Tumor will be determined as ARID1A pathologic alteration or likely pathologic alteration (Cohort I) or ARID1A wildtype by NGS (Cohort II). The number of patients in PART II cohort with clear cell or endometrioid EMCA will be capped at 33% (5 patients per cohort). Institutional pathology reports must be provided indicating at least 50% endometrioid or clear cell morphology for ovarian cancer. * Age \>= 18 * Eastern Cooperative Oncology Group (ECOG) Performance Status of =\< 2 * Prior Treatment * 1-3 prior cytotoxic therapies * NOTE: For platinum-resistant HGSOC (PART 1) may have received up to 3 prior cytotoxic therapies after developing platinum resistant disease. * Subjects with microsatellite instability- high (MSI-H) and/or mismatch repair protein deficient (dMMR) endometrioid endometrial cancer must have previously received an immune checkpoint inhibitor. * Unlimited prior hormonal therapy, targeted therapy (including immunotherapy), and/or antiangiogenic therapy will be permitted. * Washout periods (due to risk of myelosuppression): * Cytotoxic chemotherapy - 3 weeks. * Radiation therapy - 2 weeks (NOTE: patients with radiation to \> 25% of the bone marrow are NOT eligible). * Disease status: * For PART I, evaluable disease or measurable disease required. NOTE: evaluable disease: defined as disease related abnormalities on radiographic imaging that do not meet RECIST 1.1 definitions for target lesions. * For PART II, measurable disease by RECIST 1.1 is required. Patients will be required to undergo biopsy, which may be a non-target lesion but should not be the only RECIST measurable lesion. NOTE: Patients for PART II are required to undergo paired tumor biopsies. If at time of biopsy the biopsy is deemed unsafe by interventional radiology or attempted and is unsuccessful, patients may still enroll. * Hemoglobin \>= 9 g/dL (in the absence of transfusion within 28 days prior to dosing) * Absolute neutrophil count \>= 1,500 cells/mm\^3 * Platelet count \>= 100,000 cells/mm\^3 * Calculated creatinine clearance (CrCL) of \>= 50 mL/min by the Cockcroft-Gault formula * Total bilirubin =\< 1.5 x institutional upper limit of normal (ULN) (patients with known Gilbert's disease who have bilirubin level =\< 3 x ULN may be enrolled) * Aspartate aminotransferase (AST) and alanine aminotransferase (ALT) =\< 3 x institutional ULN * Patients with known history or current symptoms of cardiac disease or history of treatment with cardiotoxic agents should be New York Heart Association (NYHA) Functional Classification of class I or II. * The effects of M1774 and ZEN003694 on the developing human fetus are unknown. For this reason and because BETi agents are known to be teratogenic, women of child-bearing potential must agree to use adequate contraception (hormonal or barrier method of birth control; abstinence) prior to study entry and for the duration of study participation. Should a woman become pregnant or suspect she is pregnant while she is participating in this study, she should inform her treating physician immediately. Women of reproductive potential should use effective contraception treatment with M1774 and ZEN003694 and for at least 6 months following the last dose. Women should be advised not to breastfeed while taking M1774 and ZEN003694 and for 1 month after cessation of 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. * Patients with treated brain metastases are eligible if follow up brain imaging after central nervous system (CNS) directed therapy shows no evidence of progression, are off steroids, and are stable for at least 1 month. * The patient or a legally authorized representative must provide study-specific informed consent prior to study entry and, for patients treated in the United States (U.S.), authorization permitting release of personal health information. * PART II only: Participants must have known mutational status (wild-type or pathogenic or likely pathogenic alteration) for ARID1A by Next-Generation Sequencing. This can be determined according to local testing generated by an assay with appropriate regulatory status. * Patients must be able to swallow oral medications (capsules and tablets) without chewing, breaking, crushing, opening, or otherwise altering the product formulation. * Patients with co-morbidities: * Human immunodeficiency virus (HIV)-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. * Resolution of all toxicities of prior therapy or surgical procedures to baseline or grade 1 (except for hypothyroidism requiring medication, which must have resolved to Grade =\< 2), alopecia, and other toxicities considered clinically nonsignificant and/or stable on supportive therapy as determined by the investigator). Exclusion Criteria: * Patients who are receiving any other investigational agents. * Patients who have received prior ATR, ATM, CHK, BET, EZH2, and/or PI3K inhibitors. * History of allergic reactions attributed to compounds of similar chemical or biologic composition to ZEN003694 or M1774 used in study. * Patients taking proton pump inhibitors given decreased solubility of M1774 with increased pH. Proton pump inhibitors must be discontinued 7 days prior to initiating the trial. * Patients with corrected QT (QTc) over 450msec that does not correct with correction of electrolyte abnormalities or family history of long QT syndrome. * Patients with severe, active co-morbidity defined as follows: * No active infection requiring parenteral antibiotics. * Known hereditary diseases characterized by genetic defects of DNA repair mechanisms, including ataxia telangiectasia, Nijmegen breakage syndrome, Werner syndrome, Bloom Syndrome, Fanconi anemia, xeroderma pigmentosum, Cockayne syndrome, and trichothiodystrophy. * Pregnant and breastfeeding women are excluded from this study because ZEN003694 has the potential for teratogenic or abortifacient effects and M1774 is genotoxic in in vivo nonclinical studies. Patients who discontinue breastfeeding are eligible for enrollment and may not resume breastfeeding until 1 month off treatment. * Patients receiving any medications or substances that are strong inhibitors or inducers of CYP3A4 are ineligible. Strong inhibitors or inducers of CYP3A4 must be discontinued at least 7 days prior to the first dose of ZEN003694. Moderate inhibitors of CYP3A4 should be avoided. If alternative is not available, the use of moderate CYP3A4 inhibitors is permitted with careful monitoring and approval by study team. At the discretion of the provider, additional monitoring (labs, toxicity checks) may be implemented for use of moderate CYP3A4 inhibitors. Substrates of CYP1A2 with narrow therapeutic window also must be avoided white taking ZEN003694. Because the lists of these agents are constantly changing, it is important to regularly consult a frequently updated medical reference. As part of the enrollment/informed consent procedures, the patient will be counseled on the risk of interactions with other agents, and what to do if new medications need to be prescribed or if the patient is considering a new over-the-counter medicine or herbal product. * Patients receiving any medications or substances that are Factor Xa inhibitors are discouraged given concerns for thrombocytopenia (i.e., rivaroxaban, apixaban, betrixaban, edoxaban otamixaban, letaxaban, eribaxaban) and Factor IIa inhibitors (i.e., dabigatran). Low molecular weight heparin is allowed. If patients are not willing to switch to low molecular heparin, they must obtain approval by study team. * Serious gastrointestinal bleeding within 3 months, refractory nausea and vomiting, uncontrolled diarrhea, known malabsorption, significant small bowel resection or gastric bypass surgery, use of feeding tubes, presence of drainage gastrostomy tube, other chronic gastrointestinal disease, and/or other situations that may preclude absorption of oral medications M1774 and/or ZEN003694. * M1774 restrictions: * Patients who cannot discontinue drugs that are strong inhibitors of CYP3A4 or CYP1A2. * Patients who cannot discontinue drugs that use hMATE1 or hMATE2-K substrates.