Might help to give the general route you have in mind and for that matter the time of year. The South can be pretty cheap as long as you can stay out of trouble.
If cost is a big concern planning can make a huge difference. Often cheap campsites exist, but you have to know in advance where they are, you can't be searching for them on the bike.
Once you have a basic route you might want to ask for both local info and perhaps even try to line up some places to spend a night. You might even land a night in a real bad with a shower for free or perhaps some company for a few days on the road. Having possible help/places to stay would be a huge plus if it turns out you need to have the bike in the shop for a while like a previous poster.