DR-18 to Prevent or Treat Acute Myeloid Leukemia or Myelodysplastic Syndrome Relapse After Hematopoietic Cell Transplantation, the DR. DREAM Trial

Inclusion Criteria: * ≥ 18 years of age (no upper age limit) * The most recent HCT was from a 10/10 human leukocyte antigen (HLA)-matched related or unrelated donor (assessed at HLA-A, B, C, DR, DQ) * Evidence of blood count recovery at any time post-HCT defined as absolute neutrophil count (ANC) ≥ 0.5 x 10\^9/L for ≥ 3 consecutive days and platelets ≥ 30 x 10\^9/L (independent of granulocyte colony-stimulating factor \[G-CSF\] or platelet transfusions for 5 days). (Blood count recovery may not be sustained.) * Absent, stable or reducing immune suppression in the preceding 4 weeks without GvHD flares * Karnofsky performance status (KPS) ≥ 80% * Agrees to use a suitable method of contraception during study treatment and for 4 months after the last dose of DR-18 * Capable of providing informed consent * GROUP 1: Documented persistent or recurrent measurable residual AML or MDS after HCT, defined as bone marrow blasts \< 5% by morphology (unless suspected to be regenerative) and malignant bone marrow blasts \< 5% by flow cytometry. * Note: MRD (\< 5% malignant blasts) must be detected with flow cytometry testing at University of Washington Medical Center (UWMC)/Fred Hutchinson Cancer Center (Fred Hutch) clinical laboratory * GROUP 1: Absence of circulating malignant blasts detected by the complete blood count (CBC) * GROUP 1: Absence of extramedullary disease * GROUP 1: Post-HCT restaging never detected overt relapse or disease persistence, defined as ≥ 5% (non-regenerative) blasts by morphology or flow cytometry, or circulating malignant blasts on CBC, or extramedullary disease * GROUP 1: At least 60 days have elapsed since the HCT donor cell infusion (HCT day 0). (There is no upper limit to the time elapsed since HCT.) * GROUP 1: No Food and Drug Administration (FDA)-approved targeted therapy for the participant's AML or MDS is available, or if such therapy is available, that class of drugs previously failed for the participant or the participant was intolerant of the therapy * GROUP 2: HCT is with post-transplant cyclophosphamide or other form of in vivo or ex-vivo T cell depletion * GROUP 2: Informed consent was signed pre-HCT or latest day 28 post-HCT. (Treatment may start as early as 30 days post-HCT.) * GROUP 2-ONLY IF AML BY WORLD HEALTH ORGANIZATION (WHO) OR INTERNATIONAL CONSENSUS CLASSIFICATION (ICC) 2022 CRITERIA: Pre-HCT bone marrow shows residual leukemia by flow cytometry (MRD or overt disease with ≥ 5% blasts) * GROUP 2-ONLY IF AML BY WHO OR ICC 2022 CRITERIA: The participant had at least 1 course of intensive induction chemotherapy or 2 cycles of hypomethylating therapy with venetoclax or 4 cycles of hypomethylating therapy prior to HCT * GROUP 2-ONLY IF MDS BY WHO OR ICC 2022 CRITERIA: Any one of the following high-risk features was detected in the pre-HCT disease course: * International Prognostic Scoring System-MDS (IPSS-M) score "very high" risk * TP53 pathogenic or likely pathogenic mutation with variant allele frequency (VAF) ≥ 50% * TP53 pathogenic or likely pathogenic mutation with VAF \< 50% and complex cytogenetics or 5q or 7q cytogenetic abnormalities * Biallelic TP53 mutations * The patient had at least 1 course of intensive induction chemotherapy or 2 cycles of hypomethylating therapy with venetoclax or 4 cycles of hypomethylating therapy and the pre-HCT bone marrow evaluation still shows \> 5% malignant blasts by flow cytometry Exclusion Criteria: * Cellular immunotherapy or new targeted therapy in the 4 weeks prior to the first DR-18 injection * History of grade 3 or 4 acute GvHD after the most recent HCT * History of moderate or severe chronic GvHD after the most recent HCT * Active acute or chronic GvHD or other immunologic phenomenon (e.g., immune cytopenias, cryptogenic immunologic pneumonia) in last month requiring systemic therapy (Hydrocortisone or prednisone for adrenal insufficiency at ≤ 10 mg/day prednisone-equivalent is permitted.) * Active moderate-severe thrombotic microangiopathy (TMA) as evidenced by any of the following: \> 10 schistocytes per high-power field, or required anti-C5 or other anti-complement therapy for TMA in the prior 4 weeks, any of the following manifestations if attributed to TMA in the prior 4 weeks: hypertension, worsening or new renal insufficiency, ≥ 2+ proteinuria, hematochezia, seizure, transient or ongoing neurologic deficits * Renal insufficiency: Estimated glomerular filtration rate (eGFR) (calculated per the performing laboratory's standard formula) or measured 24 hour (hr.) creatinine clearance \< 30 mL/min * Hemodialysis in the prior 4 weeks * Major cardiac event requiring evaluation in the emergency room (ER) or hospitalization in the past 4 weeks * New York Heart Association (NYHA) class II or higher congestive heart failure (CHF) in the past 4 weeks * Uncontrolled cardiac arrhythmias, including atrial fibrillation * Left ventricular ejection fraction (LVEF) \< 35%. LVEF may be established with echocardiogram or MUGA scan, and left ejection fraction must be ≥ 35% * Liver dysfunction: aspartate aminotransferase (AST) or alanine aminotransferase (ALT) \> 5 x upper limit of normal (ULN) or bilirubin \> 3 x ULN * Active uncontrolled infection. Note: Examples of controlled infections: * Bacterial infection may be still requiring antibiotics at the time of enrollment, but clinical signs and symptoms of the infection should be improving. If the participant had bloodstream infection, negative blood cultures off antibiotics must be documented prior to initiating DR-18 treatment. For urinary tract infection, a repeat urine culture must be sterile prior to initiating DR-18. Radiographic improvement of bacterial pneumonia may lag behind clinical improvement so is not mandatory prior to DR-18 initiation * Fungal infection may be still requiring antifungal medication at the time of enrollment, but evidence of clinical response to antifungal medication (such as regression of lesions on chest CT) must be available at the time of enrollment * Asymptomatic shedding of respiratory viruses after cessation of antiviral therapy, or if not specifically treated with antiviral therapy, is permitted * Cytomegalovirus (CMV) viremia or organ infection meeting institutional criteria for CMV treatment with antiviral therapy such as ganciclovir, valganciclovir or foscarnet must be on maintenance phase of treatment or must have completed treatment and must not be in the induction treatment phase at the time of enrollment. Low-level CMV viremia not meeting institutional criteria for antiviral therapy is permitted, including low-level viremia in patients receiving CMV prophylaxis with letermovir * Any of the following: Pulmonary dysfunction requiring supplemental oxygen, even intermittently, in the past 2 weeks; corrected diffusion capacity of the lung for carbon monoxide (DLCO) or forced expiratory volume in 1 second (FEV1) \< 60% predicted; bronchiolitis obliterans syndrome; prior diagnosis of idiopathic pulmonary fibrosis; prior diagnosis of drug-induced pneumonitis; cryptogenic organizing pneumonia under active treatment * Seizure in the past 4 weeks or significant underlying neurologic disease: Study participants must not have significant active underlying neurologic disease, such as Parkinson's disease, amyotrophic lateral sclerosis, multiple sclerosis, epilepsy, prior symptomatic ischemic or hemorrhagic stroke, or transient ischemic attack, unless approved by principal investigator (PI). Peripheral neuropathy related to diabetes or prior chemotherapy is acceptable * Other medical, social, or psychiatric factor that interferes with medical appropriateness and/or ability to comply with study, as determined by the PI * Known allergic reactions to any of the components of study treatments * Concurrent use of other investigational anti-cancer agents * Peripheral blood T cell chimerism \< 40% * Pregnant or breastfeeding