This phase II trial tests the safety, side effects, and how well combination chemotherapy with fludarabine, high-dose cytarabine, granulocyte colony-stimulating factor (G-CSF), and idarubicin (FLAG-Ida) followed immediately by reduced-intensity total body radiation therapy, called total body irradiation (TBI), and donor hematopoietic cell transplant (HCT) works in treating adults age 60 and older with newly diagnosed adverse-risk acute myeloid leukemia (AML) or other high-grade myeloid cancer. Despite advances in supportive care and the approval of more than 10 new drugs since 2017, the outcomes of older adults with adverse-risk acute myeloid leukemia and other high-grade myeloid cancers remains poor. Most patients are expected to die from their cancer or the consequences of treatment-related side effects. Donor HCT is a very important part of any curative-cancer treatment for these patients. However, while accepted as standard care for decades, this treatment exposes patients to long periods of drug-induced low blood cell counts and the problems associated with low blood counts, like infections and bleeding, which are associated with significant risk of chronic side effects and death. This study will use a different approach to the upfront curative-cancer treatment of older adults with an adverse-risk AML or other high-grade myeloid cancer. This study will use intense chemotherapy followed a few days later by lower-dose TBI and donor HCT. Chemotherapy drugs, such as idarubicin, fludarabine, high-dose cytarabine work in different ways to stop the growth of cancer cells, either by killing the cells, by stopping them from dividing, or by stopping them from spreading. G-CSF helps the bone marrow make more white blood cells in patients with low white blood cell count due to cancer treatment. This approach allows effective treatment of cancer cells and overall reduction of the period of low blood cells counts. This decreases the risk for problems associated with low blood counts, such as infection and chronic side effects. Decreasing these are important for older adults who undergo HCT. This treatment strategy may improve treatment outcomes by allowing more patients to successfully undergo donor HCT and reduce the risk of low blood cell counts and the problems associated with low blood counts. Giving chemotherapy followed immediately by reduced-intensity TBI and donor HCT may be safe, tolerable and/or effective in treating adults age 60 and older with newly diagnosed adverse-risk AML or other high-grade myeloid cancer.