• DOSE-ESCALATION PHASE: Must have histologically confirmed advanced solid tumor including but not restricted to breast cancer, gastric or gastroesophageal cancer, colon cancer, endometrial cancer, salivary gland tumors, and hepatobiliary tumors

• DOSE-EXPANSION PHASE: Must have histologically confirmed advanced/metastatic gastroesophageal cancer (cohort A) or colorectal cancer (cohort B)

• DOSE-EXPANSION PHASE: Patients must have a biopsiable lesion and provide consent for on treatment biopsy

• Age \>= 18 years. Because no dosing or adverse event data are currently available on the use of AZD6738 in combination with DS-8201a in patients \< 18 years of age, children are excluded from this study

• Patients must have HER2-positive or HER2-expressing tumors determined by a Clinical Laboratory Improvement Act (CLIA)-certified laboratory. As a rule, for HER2 immunohistochemistry (IHC) scoring system trastuzumab for gastric cancer (TOGA) criteria used for gastric/gastroesophageal junction (GEJ) cancers will be employed (Note: in escalation phase, for breast cancer patients that are included, breast cancer criteria can be used). Specific requirement of HER2 status is outlined below:

‣ HER2 expression (1-3+) by IHC locally and confirmed centrally OR

⁃ HER2 expression (1-3+) by IHC tested centrally OR

⁃ HER2 amplification based on fluorescence in situ hybridization (FISH) or next generation sequencing

• Must have received at least one line of systemic chemotherapy for either locally advanced or metastatic disease and should have either progressed on this therapy or been intolerant to this therapy

• For tumors where anti-HER2 therapy is standard of care, patients must have progressed on at least 1 line of anti-HER2 therapy if eligible. For patients where DS8201a is approved as standard of care, prior treatment with DS8201a is not allowed

• Must have unresectable, advanced/metastatic disease

• Must have at least 1 measurable lesion on CT scan per Response Evaluation Criteria in Solid Tumors (RECIST) 1.1. Patient without measurable but evaluable disease are allowed for dose-escalation phase

• Must be willing and able to provide an adequate archival tumor sample available to confirm HER2 status by Central Laboratory (if local testing is used for enrollment), else must be willing and able to provide an adequate archival tumor sample for HER2 testing centrally

• Must have Eastern Cooperative Oncology Group (ECOG) performance status (PS) 0-1

• Must have life expectancy of at least 3 months

• Must have left ventricular ejection fraction (LVEF) \>= 50% within 28 days before enrollment (study drug treatment) by either an echocardiogram (ECHO) or multigated acquisition (MUGA) scan

• Must have a negative pregnancy test (if female)

• Platelets \>= 100,000/mcL (within 14 days before enrollment)

‣ No transfusions with red blood cells or platelets are allowed within 1 week prior to screening assessment

• Hemoglobin \>= 9.0 g/dL (within 14 days before enrollment)

• Absolute neutrophil count \>= 1,500/mcL (within 14 days before enrollment)

‣ No administration of granulocyte colony-stimulating factor (G-CSF) is allowed within 1 week prior to screening assessment

• Creatinine clearance \> 45/mL/min (using the Cockcroft-Gault equation) (within 14 days before enrollment)

• Aspartate aminotransferase (AST) (serum glutamic-oxaloacetic transaminase \[SGOT\])/alanine aminotransferase (ALT) (serum glutamate pyruvate transaminase \[SGPT\]) =\< 5 x institutional upper limit of normal (ULN) (within 14 days before enrollment)

• Total bilirubin =\< 1.5 x ULN if no liver metastases or \< 3 x ULN with Gilbert's Syndrome or liver metastases at baseline (within 14 days before enrollment)

• Leukocytes \>= 3,000/mcL (within 14 days before enrollment)

• Albumin \> 2.5 g/dL (GEJ patients only) (within 14 days before enrollment)

• International normalized ratio (INR) and either partial thromboplastin time (PTT) or activated (a)PTT =\< 1.5 x ULN (within 14 days before enrollment)

• Must have adequate treatment washout period before study treatment, defined as: Major surgery (\>= 4 weeks), radiation therapy (\>= 3 weeks; in case of palliative radiation \>= 2 weeks), systemic therapy (\>= 3 weeks; in case of investigational drug use \>= 2 weeks or 5 half-lives, whichever is longer)

• Patients who are human immunodeficiency virus (HIV) positive may participate IF they meet the following eligibility requirements:

‣ They must be stable on their anti-retroviral regimen, and they must be healthy from an HIV perspective

⁃ They must have a CD4 count of greater than 250 cells/mcL over the past 6 months on this same anti-retroviral regimen and must not have had a CD4 count \< 200 cells/mcl over the past 2 years, unless it was deemed related to THE CANCER AND/OR CHEMOTHERAPY-induced bone marrow suppression

∙ For patients who have received chemotherapy in the past 6 months, a CD4 count \< 250 cells/mcl during chemotherapy is permitted as long as viral loads were undetectable during this same chemotherapy

⁃ They must have an undetectable viral load and a CD4 count \>= 250 cells/mcL within 7 days of enrollment

⁃ They must not be currently receiving prophylactic therapy for an opportunistic infection and must not have had an opportunistic infection within the past 6 months. HIV-infected patients should be monitored every 12 weeks for viral load and CD4 counts

• 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

• Subjects with clinically inactive brain metastases may be included. Subjects with treated brain metastases that are no longer symptomatic and who require no treatment with corticosteroids or anticonvulsants may be included in the study if they have recovered from the acute toxic effect of radiotherapy. A minimum of 2 weeks must have elapsed between the end of whole-brain radiation therapy and study treatment

• Patients with new or progressive brain metastases (active brain metastases) or leptomeningeal disease are eligible if the treating physician determines that immediate central nervous system (CNS) specific treatment is not required and is unlikely to be required for at least 4 weeks (or scheduled assessment after the first cycle of treatment), and a risk-benefit analysis (discussion) by the patient and the investigator favors participation in the clinical trial

• 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 of the investigational regimen are eligible for this trial

• HER2 antibody conjugated to a topoisomerase 1 inhibitor agents as well as AZD6738 are known to be teratogenic; thus, women of child-bearing potential must agree to use adequate contraception (hormonal or barrier method of birth control; abstinence) prior to study entry, for the duration of study participation, and for at least 7 months (women of childbearing potential \[WOCBP\] only) after the last dose of study drug. Should a woman become pregnant or suspect she is pregnant while she or her partner is participating in this study, she should inform her treating physician immediately. Men treated or enrolled on this protocol must also agree to use adequate contraception prior to the study, for the duration of study participation, and 6 months after completion of study drug administration

• Women of non-child-bearing potential defined as pre-menopausal females with a documented tubal ligation or hysterectomy; or postmenopausal defined as 12 months of spontaneous amenorrhea (in questionable cases, a blood sample with simultaneous follicle-stimulating hormone \[FSH\] \> 40 mIU/mL and estradiol \< 40 pg/mL \[\< 147 pmol/L\] is confirmatory) are eligible. Females on hormone replacement therapy (HRT) and whose menopausal status is in doubt will be required to use one of the contraception methods outlined for women of child-bearing potential if they wish to continue their HRT during the study. Otherwise, they must discontinue HRT to allow confirmation of post-menopausal status prior to study enrollment. For most forms of HRT, at least 2-4 weeks will elapse between the cessation of therapy and the blood draw; this interval depends on the type and dosage of HRT. Following confirmation of their post-menopausal status, they can resume use of HRT during the study without use of a contraceptive method

• Male subjects must not freeze or donate sperm starting at screening and throughout the study period, and at least 6 months after the final study drug administration. Preservation of sperm should be considered prior to enrolment in this study

• Female subjects must not donate, or retrieve for their own use, ova from the time of screening and throughout the study treatment period, and for at least 7 months after the final study drug administration

• Ability to understand and the willingness to sign a written informed consent document. Participants with impaired decision-making capacity (IDMC) who have a legally-authorized representative (LAR) and/or family member available will also be eligible