First of all, were you adopted through an agency, or an attorney, or..? You can start by requesting your non-identifying information, if you haven't already. Sometimes you might get enough information from that to conduct a search yourself.
I don't know of many individuals or organizations that will do it for free; there are the costs of maintaining databases and such that force them to charge a fee. But, there are "search angels" that will help for free. Go to adopteerights.net and scroll down the page, there should be a listing of search angel resources for each state, hopefully Gershom has some for Florida on there.
If you were adopted through an agency, contact them and inquire if there is any information in your file...perhaps your mother has contacted them as well and left letters, etc. Or, perhaps they will do a search for you to get her permission for contact. Going through the agency isn't the best way, but sometimes it's the only way. It's how I had to do it to locate my mother (note: I did have to pay my agency about 575 dollars to do the search, but that was a different state, and fees vary from agency to agency and state to state). Same with if you were adopted through a lawyer...he or she may not still be in practice 44 years later, but the files should be around somewhere, do some detective work and see if you can locate who has them.
Good luck...I hate that our records are held hostage from us.
34 yo reunited adoptee