Testing the Addition of ASTX660 (Tolinapant) to the Usual Chemotherapy Treatment (Paclitaxel With or Without Bevacizumab) in Patients With Recurrent Ovarian Cancer

Inclusion Criteria: * Pathologically (histologically or cytologically) proven diagnosis of high grade epithelial ovarian, fallopian tube, or primary peritoneal cancer. Required: submission of pathology report * Patients with the following histologic cell types are eligible: * High grade serous * Endometrioid, grade 3 * Clear cell * Undifferentiated * Mixed epithelial * Carcinosarcoma * Adenocarcinoma, not otherwise specified (NOS) * Patients must be considered to have platinum-resistant or platinum-refractory recurrent ovarian cancer to be enrolled in this trial * 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 * Platinum-refractory disease is defined as progression within 30 days of completing the last dose of platinum during initial therapy. The date should be calculated from the last administered dose of platinum therapy * Patients must have evaluable disease or measurable disease defined by Response Evaluation Criteria in Solid Tumors (RECIST) version (v) 1.1. Measurable disease is defined as at least one lesion that can be accurately measured in at least one dimension (longest diameter to be recorded). Each lesion must be ≥ 10 mm when measured by CT or MRI. Lymph nodes must be ≥ 15 mm in short axis when measured by CT or MRI. Previously irradiated lesions can be considered as measurable disease only if progressive disease has been unequivocally documented at that site since radiation * 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 must have received ≥ 1 platinum-based therapy and not more than 5 prior lines of therapies. Notes: * Adjuvant/neoadjuvant therapy is counted as only 1 regimen in the absence of intervening progression. * Maintenance therapy (e.g., bevacizumab, poly adenosine diphosphate-ribose polymerase \[PARP\] inhibitor will be considered part of the preceding line of therapy \[i.e., not counted independently\]) * Therapy changed due to toxicity in the absence of progression will be considered part of the same line (i.e., not counted independently) * Hormonal therapy will not be counted as a separate line of therapy * Age ≥ 18 * Eastern Cooperative Oncology Group (ECOG) performance status ≤ 2 * Absolute neutrophil count (ANC) ≥ 1,500 cells/mm\^3 * Platelets ≥ 100,000 cells/mm\^3 * Hemoglobin ≥ 8 g/dl * Creatinine ≤ institutional upper limit of normal (ULN), OR 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 a known history or current symptoms of cardiac disease, or history of treatment with cardiotoxic agents, should have a clinical risk assessment of cardiac function using the New York Heart Association Functional Classification. To be eligible for this trial, patients should be class 2B or better * 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 * 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 * No active infection requiring parental antibiotics * No evidence of intra-abdominal abscess, abdominal/pelvic fistula, gastrointestinal perforation, gastrointestinal (GI) obstruction, and/or with drainage gastrostomy tube required. NOTE: required interval since last bowel obstruction: 30 day minimum for incomplete obstruction, resolved with conservative means; 6 months for fistula Exclusion Criteria: * Patients who have received prior weekly paclitaxel in a platinum-resistant setting * Major surgical procedure within 28 days prior to registration, or anticipation of need for major surgical procedure during the study. Note: Placement of a vascular access device, thoracentesis, and/or paracentesis will not be considered major surgery * Women who are pregnant or are unwilling to discontinue nursing * Evidence of bleeding diathesis or clinically significant coagulopathy within the past 3 months. Patients are not excluded for past or current use of anticoagulation * Uncontrolled hypertension (systolic blood pressure \[SBP\] \> 150 and/or diastolic blood pressure \[DBP\] \> 90) * Patients currently taking and unwilling/unable to discontinue the use of drugs that are known to inhibit or induce P-glycoprotein (gp)