Nab-Paclitaxel and Bevacizumab in Treating Patients With Unresectable Stage IV Melanoma or Gynecological Cancers

Inclusion Criteria: * Age \>= 18 years * Melanoma cohort only: histologic proof of surgically unresectable stage IV malignant melanoma * Melanoma cohort only: measurable disease defined as at least one lesion whose longest diameter can be accurately measured as \>= 2.0 cm with chest x-ray, or as \>= 1.0 cm 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 * Gynecologic cancer cohorts only (dose escalation and dose expansion cohorts) * Dose escalation cohort only: Histologic proof of epithelial cervical, endometrial, ovarian, fallopian, or primary peritoneal cancers; allowable histologies for cervical cancer include squamous cell carcinoma, adenocarcinoma, and mixed/adenosquamous carcinoma; allowable histologies for endometrial cancer include endometrioid, serous, clear cell, mucinous, squamous, transitional cell, undifferentiated, mixed, and carcinosarcoma (this is considered a poorly differentiated epithelial tumor); allowable histologies for ovarian, fallopian, and peritoneal cancer include serous, clear cell, endometrioid, mucinous, transitional cell, undifferentiated, mixed, and carcinosarcoma * Endometrial cancer expansion cohort only: * Histologic proof of endometrial cancer including endometrioid, serous, clear cell, mucinous, undifferentiated, mixed, and carcinosarcoma histologies * 1-3 lines of cytotoxic or immune checkpoint inhibitor therapy (not including hormonal therapy or other regimens not containing cytotoxic agents or immune checkpoint inhibitors) * If one (1) prior line of therapy, must have contained a taxane, a platinum drug, and and immune checkpoint inhibitor * If 2-3 prior lines of therapy, at least one must have contained a taxane and a platinum drug, and at least one must have contained an immune checkpoint inhibitor * Ovarian cancer expansion cohort only: * Histologic proof of ovarian cancer including high grade serous, endometrioid, clear cell, mucinous, transitional cell, undifferentiated, mixed, and carcinosarcoma histologies * 1-4 lines of cytotoxic chemotherapy (not including hormonal therapy or other non-cytotoxic regimens) * At least one prior line of chemotherapy must have contained a taxane and a platinum agent * If 1 or 2 prior lines of chemotherapy, patient's disease must be platinum-resistant * NOTE: Platinum-resistance is defined as any of the following occurring \< 183 days after the last dose of platinum-based chemotherapy: * Development of measurable disease (per Response Evaluation Criteria in Solid Tumors \[RECIST\] 1.1) * Progression of radiographic disease (per RECIST 1.1) * Increase in CA-125 level to \>= 2 x upper limit of normal (ULN) (if within normal limits \[WNL\] at the completion of platinum-based chemotherapy) * If CA-125 is used to determine the date of progression then it must be confirmed by a second CA-125 value \>= 7 days after the first level; the date of the first qualifying CA-125 is used to compute the platinum-free interval * Increase in CA-125 level to \>= 2 x nadir (if nadir \> ULN) * If CA-125 is used to determine the date of progression then it must be confirmed by a second CA-125 value \>= 7 days after the first level; the date of the first qualifying CA-125 is used to compute the platinum-free interval * If 3-4 prior lines of chemotherapy, may be platinum-resistant or platinum-sensitive * At least one prior line of cytotoxic chemotherapy must also have contained bevacizumab * Dose escalation cohort: For ovarian, fallopian tube, and peritoneal cancers only: Must meet criteria for one option below: * Platinum-resistant, defined as =\< 183 days from the date of the most recent dose of chemotherapy containing either carboplatin or cisplatin until the first evidence of cancer recurrence or progression (either symptoms directly attributable to cancer, radiographic recurrence of cancer, or cancer antigen 125 \[CA-125\] \> 70), confirmed \>= 7 days later (confirmation of elevated CA-125 may be beyond 183 days and still count as platinum-resistant) * Prior allergic reaction to carboplatin or cisplatin * Measurable disease, defined as at least one lesion whose longest diameter can be accurately measured as \>= 2.0 cm with chest x-ray, or as \>= 1.0 cm with CT scan or MRI scan; or CT component of a PET/CT; NOTE: Disease that is measurable by physical examination only is not eligible; EXCEPTION: Patients with ovarian, fallopian, or peritoneal cancer without measurable disease are eligible if two pretreatment CA125 values (documented on two occasions taken at least one week apart) are at least twice the upper limit of normal or twice the nadir value if pretreatment CA125 values never normalized. * At least one prior systematic therapy in the metastatic setting * NOTE: exception for patients with metastatic uveal or mucosal melanoma for which there are no effective/approved front line systemic treatments * 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 (PLT) \>= 100,000/mm\^3 (obtained =\< 14 days prior to registration) * Total bilirubin =\< 1.5 mg/dL 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 (obtained =\< 14 days prior to registration) * Creatinine =\< 1.5 x ULN (obtained =\< 14 days prior to registration) * Alkaline phosphatase =\< 2.5 x ULN (obtained =\< 14 days prior to registration) * Absence of proteinuria at screening as demonstrated by one of the following (obtained =\< 14 days prior to registration): * Urine protein/creatinine (UPC) ratio \< 1.0 at screening OR * Urine dipstick for proteinuria \< 2+ (patients discovered to have \>= 2+ proteinuria on dipstick urinalysis at baseline should undergo a 24 hour urine collection and must demonstrate =\< 1 g of protein in 24 hours to be eligible) * Eastern Cooperative Oncology Group (ECOG) performance status (PS) 0 or 1 * Sensory peripheral neuropathy =\< grade 1 (per Common Terminology Criteria for Adverse Events \[CTCAE\] version \[v.\] 4.0) * Motor peripheral neuropathy = grade 0 (per CTCAE v. 4.0) * Ability to understand and the willingness to sign a written informed consent document * Willing to return to enrolling institution for follow-up 2-4 weeks after treatment discontinuation * Life expectancy \>= 90 days (3 months) * Willing to provide blood samples for correlative research purposes * Negative pregnancy test done =\< 7 days prior to registration, for persons of childbearing potential only Exclusion Criteria: * Known standard therapy for the patient's disease that is potentially curative or definitely capable of extending life expectancy; EXCEPTION: For platinum-resistant ovarian cancer, because nab-paclitaxel has known benefit, patients who may benefit from standard single agent chemotherapy are also eligible to participate * Prior therapy with an angiogenesis inhibitor =\< 28 days prior to registration * No more than 3 systemic therapies (cytotoxic or immunologic) =\< 2 years prior to registration * Melanoma cohort only: Treatment with ipilimumab =\< 6 months prior to registration. * Intravenous anti-cancer therapy or investigational agents =\< 4 weeks prior to registration or non-intravenous anti-cancer therapy or investigational agents =\< 2 weeks prior to registration * Note: No washout period is required for hormone-based therapies * Uncontrolled intercurrent illness including, but not limited to: ongoing or active infection requiring systemic treatment, symptomatic congestive heart failure, unstable angina pectoris, cardiac arrhythmia, or psychiatric illness/social situations (e.g., drug addiction) that would limit compliance with study requirements * Failure to fully recover from acute, reversible effects of prior chemotherapy regardless of interval since last treatment * History or indication of brain metastases per MRI or CT at any time prior to registration * NOTE: Patients who have had primary therapy for brain metastasis (i.e. surgical resection, whole brain radiation, or SRT even if stable) are not eligible * 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 who are unwilling to employ adequate contraception * 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 * Other active malignancy =\< 3 years prior to registration. 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 for their cancer * Other medical conditions including but 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 * Immuno-compromised 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 * 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 * Clinically significant peripheral vascular disease * History of central nervous system (CNS) disease (e.g., primary brain tumor, vascular abnormalities, etc.), clinically significant stroke or transient ischemic attack (TIA) =\< 6 months prior to registration, seizures not controlled with standard medical therapy * History of hypertensive crisis or hypertensive encephalopathy * Conditions that increase the risk of venous thrombosis and/or pulmonary emboli including, but not limited to: prior history of deep venous thrombosis or pulmonary emboli, atrial fibrillation, paroxysmal atrial fibrillation, known and documented thrombophilia requiring long term anticoagulation therapy, permanent intravenous indwelling catheters, severe obesity (body mass index \[BMI\] \> 40) * Note: Patients with increased risk of venous thrombosis and/or pulmonary emboli and who are receiving anticoagulation with warfarin, low molecular weight heparin, or a direct anticoagulant are not excluded * For gynecologic cancer cohort only (dose escalation and dose expansion cohorts): Recurrent or progressive disease within 30 days of the last dose of weekly paclitaxel or nab-paclitaxel * History of inflammatory bowel disease requiring ongoing therapy * History of diverticulitis or pancreatitis =\< 6 months prior to registration * History of grade 3 or 4 bowel toxicity from immune checkpoint inhibitor =\< 12 weeks prior to registration * Invasive surgery =\< 6 weeks prior to registration, or planned elective invasive surgery during study treatment. NOTE: Patients with recent minor surgical procedures with minimal risk for wound healing complications may register =\< 6 weeks after the procedure with documented approval by the surgical team