When you enter Texas stop at their Welcome Center and request their public campground booklet. It is great!
Jan Feb March are popular months. You might call places of interest a week before you plan to arrive and just make a reservation to be sure. I definitely wouldn't plan out your whole trip. A lot can happen enroute. You might want to stay at a place longer than you planned.
We found Louisiana and Mississippi not typical snowbird places so they're not as busy.
We always used Gulf State Park at Gulf Shores, Alabama. It's very popular and if you want to stay monthly you might have a wait. They usually have short-term stay sites open as folks come and go at those.
For Florida, the popular 'warm' cities below mid-Florida are going to be busy. If you don't mind staying in the middle or northern, you'll find a site without reservations.
Have a good trip! Arizona had an awesome winter last year. Are you sure you want to leave?