Yes it’s possible. I am in a very similar situation had adverse reaction after first vaccine multiple doctors told me not to get second. I live in New York State where medical exemptions are not accepted at all even though they say they are apparently. Talk to travel agencies. I applied to multiple hospitals in New York without any luck. However I got a Pennsylvania license and multiple hospitals are taking medical exemptions in PA. I would assume Florida and Texas might be similar but you have to look around and an agency would know best. You won’t have as Many options but if you’re willing to wait and look for a while I think eventually you could find a place that would take a medical exemption.