Nab-Paclitaxel/STI-3031 Complex (AP160-Complex) for the Treatment of Advanced or Metastatic Solid Tumors

Inclusion Criteria: * Provide written informed consent * PRE-REGISTRATION * Age \>= 18 years * Willingness to provide mandatory pre-registration tissue specimen for research * At least one prior systemic therapy in the metastatic setting (adjuvant or neoadjuvant therapy not included). * NOTE: There is no upper limit to the number of prior treatment regimens * Dose Escalation Cohort Only * Patients with histologically or cytologically confirmed advanced or metastatic non-neurological solid tumors, who have no curative or life prolonging therapeutic options * Melanoma Dose Expansion Cohort Only * Histologic proof of surgically unresectable stage IV malignant melanoma * Disease progression on or after anti-PD1/PDL1 antibody-based therapy in the metastatic setting (adjuvant or neoadjuvant therapy with anti-PD1/PDL1 antibody do not count) * REGISTRATION * Tissue submitted for testing at pre-registration shows minimal level of tumor staining for PDL1 (clinical test using 22c3 immunohistochemistry) demonstrating tumor staining in \>= 1% of tumor cells * Eastern Cooperative Oncology Group (ECOG) Performance Status (PS) 0, 1 or 2 * Hemoglobin \>= 9.0 g/dL (patients may be transfused to meet hemoglobin \[Hgb\] requirement) (obtained =\< 14 days prior to registration) * Absolute neutrophil count (ANC) \>= 1500/mm\^3 (obtained =\< 14 days prior to registration) * Platelet count \>= 100,000/mm\^3 (obtained =\< 14 days prior to registration) * Total bilirubin =\< 1.5 x upper limit of normal (ULN) or direct bilirubin =\< 0.4 mg/dL (obtained =\< 14 days prior to registration) * Serum glutamic-oxaloacetic transaminase (SGOT) (aspartate aminotransferase \[AST\]) =\< 2.5 x ULN or =\< 5 x ULN in case of liver metastases (obtained =\< 14 days prior to registration) * Alkaline phosphatase =\< 2.5 x ULN or =\< 5 x ULN in case of liver metastases (obtained =\< 14 days prior to registration) * Calculated creatinine =\< 1.5 x ULN or calculated creatinine clearance \>= 45 ml/min using the Cockcroft-Gault formula for subjects with creatinine \> 1.5 ULN (obtained =\< 14 days prior to registration) * Negative serum pregnancy test done =\< 7 days prior to registration, for women of childbearing potential only * No motor peripheral neuropathy * Sensory peripheral neuropathy =\< Grade 1 (per Common Terminology Criteria for Adverse Events \[CTCAE\] 5.0) * Immune-related adverse events (irAEs) from prior treatment have returned to baseline or =\< Grade 1 * For persons of childbearing potential: agreement to remain abstinent (refrain from heterosexual intercourse) or use a contraceptive method with a failure rate of \< 1% per year during the treatment period and for 6 months after the last dose of study treatment * For person able to father a child: agreement to remain abstinent (refrain from heterosexual intercourse with a person of childbearing potential) or use contraceptive measures, and agreement to refrain from donating sperm during the treatment period and for 6 months after the last dose of study treatment * Willingness to provide mandatory blood specimens for correlative research * Willingness to provide mandatory tissue specimens for correlative research * Willing to return to enrolling institution for follow-up 2-4 weeks after treatment discontinuation * Life expectancy \>= 90 days (3 months) * Dose Expansion Cohorts Only * NOTE: Melanoma Dose Expansion cohort only planned for now. Availability to add other disease-specific dose expansion cohorts possible in future protocol amendments. * Measurable disease defined as at least one lesion whose longest diameter can be accurately measured as \>= 1.0cm with computed tomography (CT) scan or magnetic resonance imaging (MRI) scan; or CT component of a positron emission tomography (PET)/CT. * NOTE: Disease that is measurable by physical examination only is not eligible Exclusion Criteria: * Any of the following because this study involves an investigational agent whose genotoxic, mutagenic, and teratogenic effects on the developing fetus and newborn are unknown: * Pregnant persons * Nursing persons * Persons of childbearing potential {and persons able to father a child} who are unwilling to employ adequate contraception * Known standard therapy for the patient's disease that is potentially curative or definitely capable of extending life expectancy * Any anti-cancer therapy or investigational agents =\< 4 weeks prior to registration * Failure to recover from prior surgery * Failure to fully recover from acute, reversible effect of prior chemotherapy regardless of interval since last treatment * Anti-PD(L)1 antibody =\< 4 weeks prior to registration * Previous grade 4 irAEs from immune checkpoint inhibitor antibody therapy * Co-morbid systemic illnesses or other severe concurrent disease which, in the judgment of the investigator, would make the patient inappropriate for entry into this study or interfere significantly with the proper assessment of safety and toxicity of the prescribed regimens * Immunocompromised patients and patients known to be human immunodeficiency virus (HIV) positive and currently receiving antiretroviral therapy * NOTE: Patients known to be HIV positive, but without clinical evidence of an immunocompromised state, are eligible for this trial * Uncontrolled intercurrent illness including, but not limited to: * ongoing or active infection * symptomatic congestive heart failure * unstable angina pectoris * cardiac arrhythmia * or psychiatric illness/social situations that would limit compliance with study requirements * Other medical conditions including be not limited to: * History of liver disease such as cirrhosis, chronic active hepatitis, chronic persistent hepatitis or hepatitis B or C. * Active infection requiring parenteral antibiotics * Active tuberculosis or active, non-infectious pneumonitis * Evidence of interstitial lung disease * New York Heart Association class II-IV congestive heart failure (Serious cardiac arrhythmia requiring medication) * Myocardial infarction or unstable angina =\< 6 months prior to registration * Congestive heart failure requiring use of ongoing maintenance therapy for life-threatening ventricular arrhythmias * Active autoimmune disease such as Crohn's disease, rheumatoid arthritis, Sjogren's disease, systemic lupus erythematosus, or similar conditions requiring systemic therapy within the past 2 years with the use of disease modifying agents, corticosteroids, or immunosuppressants or a documented history of clinically severe autoimmune disease/syndrome difficult to control in the past * EXCEPTIONS (the following are allowed): * Vitiligo or resolved childhood asthma/atopy * Intermittent use of bronchodilators or local steroid injections * Non-immunosuppressive maintenance treatments in the setting of clinically asymptomatic disease (e.g., sulfasalazine for ulcerative colitis) * Hypothyroidism or hypoadrenalism, stable on hormone replacement, * Diabetes stable with current management * History of positive Coombs's test but no evidence of hemolysis * Psoriasis not requiring systemic treatment * Conditions not expected to recur in the absence of an external trigger * Secondary adrenal insufficiency from previous hypophysitis, currently on physiologic replacement steroid dosing only * Receiving any other investigational agent which would be considered as a treatment for the primary neoplasm * Other active malignancy =\< 3 years prior to registration. Patients must not be receiving chemotherapy or immunotherapy for another cancer. Patients must not have another active malignancy requiring active treatment * EXCEPTIONS: Non-melanotic skin cancer or carcinoma-in-situ of the cervix * NOTE: If there is a history of prior malignancy, they must not be receiving other specific treatment * NOTE: Early-stage cancer (stage 1/2, treated) should be allowed * History of myocardial infarction =\< 6 months, or congestive heart failure requiring use of ongoing maintenance therapy for life-threatening ventricular arrhythmias * Active central nervous system (CNS) metastasis * NOTE: Patients with prior brain metastases that are asymptomatic without corticosteroid use and stable or improved \>= 30 days after treatment with surgery or radiation are not excluded * Corticosteroid use =\< 14 days prior to registration. NOTE: Patients must be off systemic corticosteroids for at least 2 weeks prior to registration. This includes oral or IV route of administration. Patients on chronic corticosteroids for adrenal insufficiency or other reasons may enroll if they receive less than 10 mg/day of prednisone (or equivalent). Patients receiving inhaled or intranasal or intraarticular steroids are not excluded * EXCEPTIONS: Patients requiring steroid premedication for radiology contrast allergy are not excluded