ASTX727, Venetoclax, and Gilteritinib for the Treatment of Newly Diagnosed, Relapsed or Refractory FLT3-Mutated Acute Myeloid Leukemia or High-Risk Myelodysplastic Syndrome

Inclusion Criteria: * Diagnosis: * Phase I cohort: Adults \>= 18 years with relapsed/refractory FLT3-mutated AML or myelodysplastic syndrome (MDS) that is intermediate-2 or high-risk by the International Prognostic Scoring System * Phase II cohort A: Adults \>= 18 years with newly diagnosed FLT3-mutated AML. Patients should meet the following criteria: * Confirmed newly diagnosed AML with FLT3 mutation * Ineligible for induction therapy defined as * Either age \>= 75 * Or 18-74 with at least one comorbidity (congestive heart failure \[CHF\] requiring therapy or ejection fraction \[EF\] =\< 50%, diffusion capacity of the lung for carbon monoxide \[DLCO\] =\< 65% or forced expiratory volume in 1 second \[FEV1\] =\< 65%, or Eastern Cooperative Oncology Group \[ECOG\] 2 or 3, or other significant co-morbidity precluding use of cytotoxic chemotherapy as approved by the principal investigator (PI) * Phase II cohort B: Adults \>= 18 years with relapsed/refractory FLT3-mutated AML or MDS that is intermediate-2 or high-risk by the International Prognostic Scoring System who have received 1 prior therapy * For all cohorts, patients with either FLT3-ITD or FLT3 D835 mutations will be eligible * Performance status =\< 3 (Eastern Cooperative Oncology Group \[ECOG\] scale) * Total serum bilirubin =\< 2.5 x upper limit of normal (ULN), unless due to Gilbert's syndrome, hemolysis or the underlying leukemia approved by the PI * Alanine aminotransferase (ALT) or…