You do not have to change your name on your deed prior to refinancing. Not changing your name on the deed until you refinance will not change your homestead or taxes. When you do refinance, the mortgagor on the new mortgage will be "Mary Jones, a married woman, formerly known as Mary Smith, joined by her husband Peter Jones." Your husband will be joining in the mortgage only because he is required to by the Florida Constitution.
Disclaimer: This answer is provided for informational purposes only, does not constitute legal advice, and does not create an attorney-client relationship. Actual legal advice can only be provided after completing a comprehensive consultation in which all of the relevant facts are discussed and reviewed.