The wife and I did the same trip you are proposing last year. We made it as far as the Atlanta area in the Fall, and flew home for the holidays. Back in Spring and are currently working our way around the Gulf.
I called quite a few storage areas, primarily in the north-eastern suburbs of Atlanta. Ended up storing my 34' rig at CubeSmart in Lawrenceville. Most of their storage are small garage-like units, but they also have a gravel lot for RVs, boats, trucks etc. The site is completely fenced with key-pad locked entrance. Cost was about $110 per month.
CubeSmart is a nation-wide outfit, so their phone (855-546-8305) goes to a central office dispatcher that will check space availability at that site, and possibly other sites in the area. The manger at the Lawrenceville site was Maria. The dispatcher will give you the local number to talk with her about the details.
My choice of north-eastern Atlanta suburbs was based on having family in that area to stay with while we readied the rig for storage. I had tires installed, engine servicing and winterizing done locally. We packed up a couple boxes and shipped them home via FedEx also. The Atlanta airport is south of the city, so our commute was about 45 min to get there. We picked up a one-way rental car from Lawrenceville to the airport when we left.
Good luck with your trip ... you'll love it!
Karl and Dawn
2018 Tiffin Open Road 32SA
2013 Fiat 500C Toad