Testing the Combination of the Anticancer Drugs Trastuzumab Deruxtecan (DS-8201a) and Azenosertib (ZN-c3) in Patients With Stomach or Other Solid Tumors

Inclusion Criteria: * In the dose escalation, patients must have a histologically documented locally advanced, unresectable, or metastatic solid tumor that has progressed following at least one prior line of treatment in the metastatic setting or has no satisfactory alternative treatment option and all of the following: * HER2 expression by immunohistochemistry (IHC) (1+, 2+, or 3+) or HER2 amplification by in situ hybridization (ISH) or next generation sequencing (NGS) (on any Clinical Laboratory Improvements Amendments \[CLIA\] platform on tissue), AND * T-DXd (DS-8201a)-naive disease * In the dose expansion, patients must have histologically documented locally advanced, unresectable or metastatic gastric or gastroesophageal junction (GEJ) cancer that has progressed following at least one prior line of treatment in the metastatic setting and have all of the following: * HER2 expression by IHC (1+, 2+, or 3+) or HER2 amplification by ISH or NGS (on any CLIA platform on tissue), AND * T-DXd (DS-8201a)-naive disease * Received prior trastuzumab-based treatment, if eligible for such treatment * For the dose escalation and dose expansion, patients can have evaluable or measurable disease * Potential trial participants should have recovered from clinically significant adverse events (AEs) of their most recent therapy/intervention prior to enrollment * Age ≥ 18 years. Because no dosing or AE data are currently available on the use of T-DXd (DS-8201a) in combination with azenosertib (ZN-c3) in patients \< 18 years of age, children are excluded from this study * Eastern Cooperative Oncology Group (ECOG) performance status of ≤ 1 (Karnofsky ≥ 70%). Both T-DXd (DS-8201a) and azenosertib (ZN-c3) have fatigue as an adverse effect. Due to the overlapping adverse effect, the performance status cannot be less restrictive * Absolute neutrophil count ≥ 1.5 × 10\^9/L (within 7 days of study treatment initiation) * No transfusions with red blood cells or platelets are allowed within 1 week prior to screening assessment * No administration of granulocyte colony-stimulating factor is allowed within 1 week prior to screening assessment * Hemoglobin \> 9.0 g/dL (within 7 days of study treatment initiation) * No transfusions with red blood cells or platelets are allowed within 1 week prior to screening assessment * No administration of granulocyte colony-stimulating factor is allowed within 1 week prior to screening assessment * Platelets ≥ 100 × 10\^9/L (within 7 days of study treatment initiation) * No transfusions with red blood cells or platelets are allowed within 1 week prior to screening assessment * No administration of granulocyte colony-stimulating factor is allowed within 1 week prior to screening assessment * Total bilirubin ≤ 1.5 institutional upper limit of normal (ULN). Documented Gilbert syndrome is allowed if total bilirubin is ≤ 3 × institutional ULN (within 7 days of study treatment initiation) * No transfusions with red blood cells or platelets are allowed within 1 week prior to screening assessment * No administration of granulocyte colony-stimulating factor is allowed within 1 week prior to screening assessment * Aspartate aminotransferase (AST \[serum glutamic-oxaloacetic transaminase (SGOT)\])/alanine aminotransferase (ALT \[serum glutamate pyruvate transaminase (SGPT)\]) ≤ 3 × institutional ULN. In the presence of liver metastases, AST or ALT up to 5 × institutional ULN is permitted (within 7 days of study treatment initiation) * No transfusions with red blood cells or platelets are allowed within 1 week prior to screening assessment * No administration of granulocyte colony-stimulating factor is allowed within 1 week prior to screening assessment * Measured of calculated creatinine clearance (CrCl) ≥ 60 mL/min (CrCl should be calculated per institutional standard; glomerular filtration rate can also be used in place of CrCl) ≥ 60 mL/min for patients with creatinine levels \> 1.5 x institutional ULN (within 7 days of study treatment initiation) * No transfusions with red blood cells or platelets are allowed within 1 week prior to screening assessment * No administration of granulocyte colony-stimulating factor is allowed within 1 week prior to screening assessment * International normalized ratio/prothrombin time and activated partial thromboplastin time ≤ 1.5 × institutional ULN (within 7 days of study treatment initiation) * No transfusions with red blood cells or platelets are allowed within 1 week prior to screening assessment * No administration of granulocyte colony-stimulating factor is allowed within 1 week prior to screening assessment * Patients must have left ventricular ejection fraction (LVEF) ≥ 50% by either an echocardiogram (ECHO) or multigated acquisition (MUGA) scan within 28 days before enrollment * Human immunodeficiency virus-infected patients on effective anti-retroviral therapy with undetectable viral load within 6 months are eligible for this trial * For patients with evidence of chronic hepatitis B virus (HBV) infection, the HBV viral load must be undetectable on suppressive therapy, if indicated * Patients with a history of hepatitis C virus (HCV) infection must have been treated and cured. For patients with HCV infection who are currently on treatment, they are eligible if they have an undetectable HCV viral load * Patients with treated brain metastases are eligible if follow-up brain imaging after central nervous system (CNS)-directed therapy shows no evidence of progression * Patients with new or progressive brain metastases (active brain metastases) or leptomeningeal disease are eligible if the treating physician determines that immediate CNS-specific treatment is not required and is unlikely to be required during the first cycle of study treatment * Patients with a prior or concurrent malignancy whose natural history or treatment does not have the potential to interfere with the safety or efficacy assessment