• 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).