Testing the Use of Ado-Trastuzumab Emtansine Compared to the Usual Treatment (Chemotherapy With Docetaxel Plus Trastuzumab) or Trastuzumab Deruxtecan for Recurrent, Metastatic, or Unresectable HER2-Expressing Salivary Gland Cancers

Inclusion Criteria: * Pathologically (histologically or cytologically) proven diagnosis of HER2-positive OR HER2-low expressing recurrent/metastatic salivary gland cancer (SGC) * HER2-positive cohort: * Note: The majority of HER2-positive SGCs are salivary duct carcinoma (SDCs), but to a lesser extent, other SGC subtypes can be HER2-positive (e.g., adenocarcinomas, mucoepidermoid carcinomas, etc.) and are eligible to be included on the study. Additionally, pathologists may sign out SDCs under other descriptors (e.g., ex-pleomorphic adenoma, adenocarcinoma), and these would be eligible if they are HER2-positive. * Note: HER2 evaluation based on local site immunohistochemistry (IHC), fluorescent in-situ hybridization (FISH), or local/commercial next-generation sequencing (NGS) is required. Any one of the following criteria observed in a primary tumor or metastasis would meet the study definition for "HER2-positive": * Immunohistochemistry (IHC) (3+) per the College of American Pathologists (CAP) breast cancer guidelines * Gene amplification by FISH (HER2/CEP17 ratio \>= 2.0) * Gene amplification by NGS (fold change \>= 2) * HER2-low expressing cohort: * Note: Local HER2 evaluation by immunohistochemistry (IHC) or fluorescent in-situ hybridization (FISH) is required. Any one of the following criteria observed in a primary tumor or metastasis would meet the study definition for "HER2-low": * IHC 1+ per the College of American Pathologists (CAP) breast cancer guidelines * IHC 2+ without evidence of amplification by FISH * Patients with unresectable disease who are not candidates for curative surgery or radiation OR recurrent OR metastatic disease that is evident on radiologic imaging * 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 therapy * HER2-positive cohort: Measurable or non-measurable disease by the RECIST v1.1 criteria. HER2-low expressing cohort: Measurable disease by the RECIST v1.1 criteria * History/physical examination within 30 days prior to registration * The following imaging within 60 days prior to registration: * CT or MRI of the neck (diagnostic quality with contrast, unless contraindicated) AND * CT scan of the chest (diagnostic quality with contrast, unless contraindicated) AND * If clinically indicated, CT or MRI of the abdomen and pelvis (diagnostic quality with contrast, unless contraindicated) * Age \>= 18 * Left ventricular ejection fraction (LVEF) \>= 50% assessed by echocardiogram or multigated acquisition (MUGA) scan within 30 days prior to registration * Zubrod (Eastern Cooperative Oncology Group \[ECOG\]) Performance Status of 0-2 within 14 days prior to registration * Absolute neutrophil count (ANC) \>= 1,500 cells/mm\^3 (within 14 days prior to registration) * Platelets \>= 100,000 cells/mm\^3 (within 14 days prior to registration) * Hemoglobin \>= 9.0 g/dL (within 14 days prior to registration) * HER2-positive cohort: Note: The use of transfusion or other intervention to achieve hemoglobin \[Hgb\] \>= 9.0 g/dL is acceptable * HER2-low expressing cohort: Note: Transfusion (red blood cell or platelet) or granulocyte-colony stimulating factor (granulocyte colony-stimulating factor \[G-CSF\]) is not allowed * Serum creatinine =\< 1.5 x upper limit of normal (ULN) OR calculated creatinine clearance (CrCl) \>= 30 mL/min by the Cockcroft-Gault formula (within 14 days prior to registration) * HER2-positive cohort: Total bilirubin =\< 1.5 x ULN (within 14 days prior to registration) (Not applicable to patients with known Gilbert's syndrome) (within 14 days prior to registration) * HER2-positive cohort: Aspartate aminotransferase (AST) and alanine aminotransferase (ALT) =\< 1.5 x ULN (within 14 days prior to registration) * HER2-low expressing cohort: Total bilirubin ≤ 1.5 x ULN if no liver metastases; or \< 3 x ULN in the presence of documented Gilbert's Syndrome or liver metastases (within 14 days prior to registration) (within 14 days prior to registration) * HER2-low expressing cohort: AST and ALT ≤ 3 x ULN if no liver metastases; or \< 5 x ULN with liver metastases (within 14 days prior to registration) * HER2-low expressing cohort: Serum albumin ≥ 2.5 g/dL (within 14 days prior to registration) * Known human immunodeficiency virus (HIV) infected patients on effective anti-retroviral therapy with undetectable viral load within 6 months prior to registration are eligible for this trial. Testing is not required for entry into protocol * For patients with known evidence of chronic hepatitis B virus (HBV) infection, the HBV viral load must be undetectable on suppressive therapy, if indicated * Note: Known positive test for hepatitis B virus surface antigen (HBV sAg) indicating acute or chronic infection would make the patient ineligible unless the viral load becomes undetectable on suppressive therapy. Patients who are immune to hepatitis B (anti-hepatitis B surface antibody positive) are eligible (e.g., patients immunized against hepatitis B) * For patients with a known history of hepatitis C virus (HCV) infection, they 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 * Note: Known positive test for hepatitis C virus ribonucleic acid (HCV RNA) indicating acute or chronic infection would make the patient ineligible unless the viral load becomes undetectable on suppressive therapy * Negative urine or serum pregnancy test (in persons of childbearing potential) within 14 days prior to registration. Childbearing pote