Take a look at the back of your license. It says "Direct Inquiries To: Florida Department of State, Division of Licensing, Post Office Box 6687, Tallahassee, Florida 32314-6687 850/488-5381." I had a problem with mine and was perplexed as to how to correct it from California when I noticed the number on the back of the license and gave them a call.
They are very nice, friendly people who will help you change the address on your license and your status to a non-resident, most likely, but not necessarily, over the phone. They can even send you a new license with your new address on it. My transaction took about five minutes and the new license showed up in two weeks.
They will require a new fingerprint card every time you renew as a non-resident, and the fee will be a bit higher, but it's not really a big deal. Changing to a non-resident status may require the fingerprint card now, to make sure that they are talking with the actual licensee.