Busulfan, Fludarabine Phosphate, and Anti-Thymocyte Globulin Followed By Donor Stem Cell Transplant and Azacitidine in Treating Patients With High-Risk Myelodysplastic Syndrome and Older Patients With Acute Myeloid Leukemia

Inclusion Criteria: * Meets one of the following sets of criteria: * Myelodysplastic syndromes (MDS): * Disease with high-risk features (found either at diagnosis or before initiation of cytotoxic therapy), defined as one of the following: * International prognostic scoring system (IPSS) risk \>= intermediate-2 * Refractory anemia with excess blasts by French-American-British (FAB) classification * High-risk cytogenetics (either complex or -7) * Less than 10% bone marrow blasts as determined by bone marrow biopsy within the past 4 weeks (reduction in marrow blast percentage may be achieved with chemotherapy or other therapy) * Less than 75 years old * Acute myeloid leukemia (AML): * No FAB M3 * No acute leukemia following blast transformation of prior chronic myelogenous leukemia or other myeloproliferative disease * Patients with preceding MDS or treatment-related AML are eligible * Prior central nervous system (CNS) involvement is allowed provided the disease is in remission at transplantation * Morphologic complete remission (leukemia-free state) is defined as meeting all of the following criteria: * Bone marrow blasts \< 5% (as determined by bone marrow within the past 4 weeks), but without requirement for normal peripheral blood counts * No extramedullary leukemia * No blasts in peripheral blood * Achieved complete remission (CR) after no more than 2 courses of induction chemotherapy * Patients treated with azacitidine or decitabine who achieve a leukemia-free state are eligible (may have required up to 4 courses of therapy to reach this status) * Age 60 to 74 years * Donors must meet the following criteria: * One of the following: * HLA-identical sibling (6/6) by serologic typing for class (A, B) and low-resolution molecular typing for class II (DRB1) * Matched unrelated donor (8/8) by high-resolution molecular typing at HLA-A, -B, -C, and DRB1 * No syngeneic donors * Eastern Cooperative Oncology Group (ECOG) performance status 0-2 * Calculated creatinine clearance ≥ 40 mL/min * Bilirubin \< 2 mg/dL OR bilirubin 2-3 mg/dL provided direct bilirubin is normal * Aspartate aminotransferase (AST) \< 3 times upper limit of normal * Diffusing capacity of the lung for carbon monoxide (DLCO) \> 40% with no symptomatic pulmonary disease * Left ventricle ejection fraction (LVEF) \>= 30% by echocardiogram (ECHO) or multigated acquisition (MUGA) * Not pregnant or nursing * Negative pregnancy test * Fertile patients must use effective contraception * No uncontrolled diabetes mellitus or active serious infections * No known hypersensitivity to E. coli-derived products, azacitidine, or mannitol * No human immunodeficiency virus (HIV) infection or active hepatitis B or C * Prior azacitidine or decitabine allowed * No patients who progressed from MDS to AML during treatment with azacitidine or decitabine * At least 4 weeks since prior deoxyribonucleic acid (DNA)-hypomethylating chemotherapy, radiotherapy, and/or surgery * No more than 2 courses of consolidation therapy before transplantation (for patients with AML) * Any consolidation regimen that does not require transplantation can be used * No more than 6 months from documentation of morphologic CR to transplantation