Jane - sounds great!
I was living in a high income tax state prior to an expat assignment. I found I needed to actively establish residence in another state for NY to no longer consider me a resident - but (in my case anyway) no one really checked to see how much time I spent there. I ended up finding two other guys in the same situation and renting a small apartment in Florida which we spent only a handful of days in over the space of three years.
In our case it turned out to be cheaper than state taxes on our income, and it was somewhat convenient to have a place to crash in the US when we wanted to.
Your situation is different, of course. I believe the real challenge is not maintaining FL (or whatever income-tax free state) residence, but convincing GA you are no longer a resident of that state. That might be accomplished just by 'moving' to FL temporarily after you've sold your GA residence.
If it were me, I'd sell the GA place, move my goods into storage in northern FL (its probably real cheap there away form the water at least), then rent an apartment for a few months (with or without actually living there). After that it would seem to me one could leave the country as a FL resident. As you probably know there are places in the sticks in northern FL you could rent a place for a song.
Having said all that, please take the others advice and see what the other perpetual travelers have done.