I'm not sure which law governs your employment agreement, since the employer could be out of state, and your agreement could recite that it's governed by some law other than Florida's. But assuming it's Florida law (and since I'm not admitted to practice in Florida), only an attorney licensed to practice there can advise you properly. Specifically, I believe Florida limits the scope of what non-compete agreements are enforceable, based on the time of the restriction, the employer's line of business and that the restriction is reasonably necessary to protect the employer's legitimate business interests. But you would, as you suspect, be prudent not to sign the termination agreement which includes the non-compete in order to avoid being bound by it, until you consult a Florida attorney.
THE INFORMATION PRESENTED HERE IS GENERAL IN NATURE AND IS NOT INTENDED, NOR SHOULD IT BE CONSTRUED, AS LEGAL ADVICE. THIS POSTING DOES NOT CREATE ANY ATTORNEY-CLIENT RELATIONSHIP WITH THE AUTHOR (WHO IS ONLY ADMITTED TO PRACTICE IN THE STATES OF CALIFORNIA AND NEW YORK). FOR SPECIFIC ADVICE ABOUT YOUR PARTICULAR SITUATION, CONSULT YOUR ATTORNEY.