Gene-Modified T Cells With or Without Decitabine in Treating Patients With Advanced Malignancies Expressing NY-ESO-1

Inclusion Criteria: * Patients with solid tumors as described below: * Inoperable or metastatic (advanced) melanoma: * Has received, is intolerant, or refused a CTLA-4 inhibitor (ipilimumab) or a PD-1 inhibitor (nivolumab or pembrolizumab) as monotherapy and/or a combination of ipilimumab and nivolumab * Has received or is intolerant of a BRAF inhibitor or the combination of BRAF and MEK inhibitors for BRAFv600 mutant melanoma and a PD-1 inhibitor as monotherapy or in combination * Inoperable or metastatic (advanced) ovarian, primary peritoneal or fallopian tube carcinoma: * Has received platinum containing chemotherapy and has platinum refractory or resistant disease that has progressed on second line therapy * If platinum sensitive disease, should have received \>= 2 lines of chemotherapy * May have received PARP inhibitors, bevacizumab or other targeted VEGF inhibitor therapy * Inoperable or metastatic (advanced) synovial sarcoma: * Should have received and progressed on \>= two lines of systemic therapy * Subjects with other histologies: * Must have previously received two lines of systemic standard care (or effective salvage chemotherapy regimens) for metastatic disease, if known to be effective for that disease, and have been deemed either non-responders (progressive disease) or have recurred * For cohorts 1, 2 and 3 only: Patient's tumor must be positive by histological or molecular assay for NY-ESO-1, according to the screening algorithm; historical results may be used * For cohort 4, NY-ESO-1 results will be noted but NY-ESO-1 positivity is not required for eligibility * Human leukocyte antigen (HLA)-A\*0201 (HLA-A2.1) positivity by molecular subtyping (blood test or buccal swab, historical documentation acceptable) * Age \>= 18 years old, (Cohort 4: Age \>= 12 years old) * Life expectancy greater than 3 months assessed by a study physician * Have been informed of other treatment options * A minimum of one measurable lesion defined as: * Meeting the criteria for measurable disease according to irRECIST criteria * For patients with skin metastases, lesions selected as non-completely biopsied target lesion(s) that can be accurately measured and recorded by color photography with a ruler to document the size of the target lesion(s) * No restriction based on prior treatments but at least 4 weeks from prior immunotherapy, or prior investigational agents. Note: Patients who have suffered \>grade 2 irAEs during previous checkpoint inhibitor therapy should be excluded. * Eastern Cooperative Oncology Group (ECOG) performance status (PS) 0 or 1 * Must have adequate venous access for apheresis * Women of childbearing potential and men must agree to use effective methods of birth control for the duration of the study and 6 months after; methods for acceptable birth control include: condoms, diaphragm or cervical cap with spermicide, intrauterine device, and hormonal contraception; it is recommended that a combination of two methods be used * Leukocytes: \>= 3,000/mcl * Absolute neutrophil count: \>= 1,000/mcl * Platelets: \>= 100,000/mcl * Total bilirubin: =\< 1.5 upper limit of normal (ULN) * Aspartate aminotransferase (AST) (serum glutamic oxaloacetic transaminase \[SGOT\])/alanine aminotransferase (ALT) (serum glutamate pyruvate transaminase \[SGPT\]): =\< 2.5 x institutional upper limit of normal * Creatinine: =\< 2 X ULN; if creatinine \> 2 X ULN, creatinine clearance must be \> 60 ml/min * Must be willing and able to accept the leukapheresis procedure * At screening, must have tissue available for NY-ESO-1 testing (if not previously performed) or be willing and able to undergo a fresh tissue biopsy * Patient must understand the investigational nature of this study and sign an Independent Ethics Committee/Institutional Review Board approved written informed consent form prior to receiving any study related procedure * Participant must agree to and arrange for a caregiver (age \>= 18 years old) available 24 hours a day/ 7 days a week and arrange for lodging within 45 minutes drive to Roswell Park and transportation for a period of time after discharge from the hospital; the exact amount of time will depend on the individual status as determined by the treating physician * ELIGIBILITY CRITERIA FOR A SECOND TRANSGENIC T CELL INFUSION Subjects in whom disease control was observed after the T cell infusion may be eligible for a second transgenic T cell infusion if the subject meets the following criteria: * Patient attained confirmed disease control (either complete remission \[CR\], partial remission \[PR\] or stable disease \[SD\]) after the first transgenic T cell infusion * Patient still meets the eligibility criteria above * A second T cell infusion is discussed and the patient agrees to receive it * Either cryopreserved extra cells from the first T cell product is available or cryopreserved autologous peripheral blood mononuclear cell (PBMC) is available for the manufacture of a second transgenic T cell product Exclusion Criteria: * Previously known hypersensitivity to any of the agents used in this study * Currently receiving any other investigational agents. Note: Patients who have suffered \>grade 2 irAEs during previous checkpoint inhibitor therapy should be excluded. * 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; electrocardiogram (EKG) will be done at screening; cardiac stress test will be done as clinically indicated, the specific test to be chosen at the discretion of the treating physician. * History of severe autoimmune disease requiring steroids or other immunosuppressive treatments * History of inflammatory bowel disease, celiac disease, or other chronic gastrointestinal conditions associated with diarrhea or bleeding, which in the investigator's opinion would place the patient at an increased risk for adverse effect or current acute colitis of any origin; treated cases with no active disease are eligible * Potential requirement for systemic corticosteroids or concurrent immunosuppressive drugs based on prior history or received systemic steroids within the last 4 weeks prior to enrollment (inhaled or topical steroids at standard doses or isolated use of steroids as premedication for medical procedures to minimize allergic reaction \[e.g. computed tomography (CT) scan dye\] are allowed) * Known active infection with human immunodeficiency virus (HIV), hepatitis B, hepatitis C or cytomegalovirus (CMV) * Known cases of clinically active brain metastases (brain magnetic resonance imaging \[MRI\] as clinically indicated); prior evidence of brain metastasis successfully treated with surgery or radiation therapy will not be exclusion for participation as long as they are deemed under control at the time of study enrollment * Dementia or significantly altered mental status that would prohibit the understanding or rendering of compliance with the requirements of this protocol even with caregiver support * Pregnancy or breast-feeding; female patients must be surgically sterile or be postmenopausal for two years, or must agree to use effective contraception during the period of treatment and 6 months after; all female patients with reproductive potential must have a negative pregnancy test (serum/urine) within 48 hours from starting the conditioning chemotherapy * Lack of availability of a patient for immunological and clinical follow-up assessment