Free COVID-19 Testing and Flu Shots - Odessa American: Local News by The City of Odessa in partnership with 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. Lining up starts at 8:30 a.m. Monday: UTPB Park. 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. Residents must show proof of residency and provide good contact information (phone number and email) in order to get results.