When we were moving to Fl. from Pa. went to the state office building in Pittsburgh with the certificate they give you when your born & our drivers licenses - filled out a few papers and paid I believe $20 for both of us. I got the birth certificates in a few days in the mail.
Florida is very very particular about you being able to prove you are who you say. If , like me , you have been married more than once, you have to bring all your supporting paperwork. I had my birth certificate, 1st marriage certificate, first husband's death certificate, second marriage certificate and my USA passport just for good measure.They had to verify it all - we were there for a very long time, then in the end they decided my passport was enough.