It really depends on when and which part of the South you want. Most of it is warm and sunny. If you're looking at beaches, go before or after the busy season, which start with spring breaks. You can get much cheaper hotel rooms or rentals in the "off" season. I always enjoy Florida in late April. We go to Destin and the Gulf of Mexico is very nice in April and May. It's not too hot and there aren't a lot of tourists.
If you're looking for fun places to visit, I'm in Nashville and it's listed as on the the "it" places to visit right now. Lots of music, not just country, and fabulous restaurants. Again, go before the summer. Our spring weather is normally very nice but the Country Music Festival is in the summer and rates go up.
There are a ton of places to visit that are just very nice to see, i.e. Savannah, Charleston, etc. Atlanta is always a great town to visit. Austin, Texas has a great music festival and their food scene is thriving but that wouldn't help with the shoestring. I don't know if New Orleans is ever in the "off" season but if you don't go around Mardi Gras it's probably better priced.
I assume you wanted the South and not Southwest. Someone else will have to help you with that. I'm don't know where you're from but, if you come the the South, be prepared for everyone to talk to you and ask you where you're from. You will also be "ma'am"d alot!! We don't believe in strangers and will not only tell you what we did today, we'll ask, "How's your mama and them?" Which sounds like, "How's ya mama an' 'em?"