Weems Medical Center East. 110 NE 5th Street, Carrabelle (850) 697-2345. Open Monday through Friday. 8:00am to 4:30pm (closed 12:00pm to 1:00pm) Saturday from 8:00am to 12:00pm. Weems Memorial Hospital Emergency Room. 135 Avenue G, Apalachicola. Open 24 hours a day, 7 days a week..