Interesting you're having a problem. I "live" at the same address as you, but I transferred my address at the GCS DMV office and had no problem. I once got my Florida
at the DMV in Marathon (in the keys) and had a Georgia
address on it. I explained to the clerk that I lived on a boat and would be happy to furnish the DMV with my lat/long, but if they wanted to reach me by mail they would have to use my Georgia
address. They acquiesced.
I have to admit I really enjoyed having a Florida license
with a Georgia address. I sort of regret that I was never stopped by the gendarmes during that period. Only sort of.