Taladegib, Paclitaxel, Carboplatin, and Radiation Therapy in Treating Patients With Localized Esophageal or Gastroesophageal Junction Cancer

Inclusion Criteria: * Histologically or cytologically confirmed adenocarcinoma of the esophagus or gastroesophageal junction (EAC) * Localized EAC and its baseline clinical stage determined as: T2-T3N0 or T1-3N positive (+); imaging studies suspicious for metastases must be followed with a negative biopsy before a patient can enter the study * Patients with malignant celiac nodes are eligible if the primary lesion is in the mid-thoracic or distal thoracic esophagus or it is involving the gastroesophageal junction * Tumor must have labeling index of \>= 5% of the nuclear Gli-1 (integral biomarker) performed in the MD Anderson Cancer Center Clinical Laboratory Improvement Amendment (CLIA) laboratory for patient to be eligible in this trial (if enough archival tissue is not available to determine labeling index, patient must agree to a biopsy to be eligible for the study) * Tumor may not extend \> 4 cm below the gastroesophageal junction * Eastern Cooperative Oncology Group (ECOG) performance status 0 or 1 * All patients must be willing to provide research tumor tissue for biomarker studies at baseline, from archival tumor tissue or through endoscopy if sufficient archival tissue is not available; all patients must also allow biomarker studies on the tissue obtained through surgery to remove the primary cancer * Phase II only: patients volunteering for the Phase II part of the protocol must be willing to undergo a research endoscopy for tissue collection on day 8 (+/- 2 days) f…