Free Flu Shots and COVID testing - Odessa American: Local News by The City of Odessa in partnership with Medical Center Health System and Odessa Regional Medical Center are offering free COVID-19 testing and free flu shots to City of Odessa and Ector County residents. Free Mobil COVID Testing has been scheduled starting at 10 a.m. at the following dates and locations. Line up starts at 8:30 a.m. Friday: Ector County Coliseum. Dec. 21: Central Park. Dec. 28: Sherwood Park. Dec. 30: UTPB Park. Only 200 tests will be available, at each event, for City of Odessa and Ector County residents. Itâs first come, first serve.