Hi Kelly. Every state has different estate laws. It's best to check with a Florida attorney on the rules that would apply.
You can certainly put your name on the title now and/or refinance with a new mortgage under both your names.
You do NOT have to be on the mortgage, but you must be on the title to have any rights to the property. This is NOT a legal advice - I am not an attorney. You should contact a lawyer for a legal advice as every state has their own property laws.
Check with the settlement company in FL to see what the laws are there. My guess is that if you aren't on the title and he has no will, it will go to his heirs. The settlement company can advise you on how to add your name, as well.