A trip you may want to consider is a whitetail hunt in the Texas Hill Country. That's roughly considered the area around and SW of San Antonio.
http://en.wikipedia.org/wiki/Texas_Hill_Country There is no public land, but "cull hunts" can be pretty cheap, by Texas standards.
While the deer bodies are small and the racks aren't typicaly the big "trophy size" racks of "South Texas", you can get on some ranches for a fairly reasonable price, deer are very plentiful and there's a really good chance of seeing some really nice bucks, by most standards.
For bowhunting, it's one of the few places that you can still actually "still hunt" for whitetails. Most of the eastern USA land ownership is so fragmented that you can't go far without hitting a property line and most public forest land is too old growth for successful "still hunting".
Not so in the Texas Hill Country. With the hills and valleys, cactus, evergreen brush and live oak, you can spend and entire day "still hunting" and have a really good chance of seeing a ton of deer that may offer a shot opportunity.
Years back, I went on a hunt around Sonora, Texas. I had a couple days where I saw literally 30-50 bucks in a single day. Most were small, but a couple were dandy bucks.
Now, things have changed since then. Prices for large racked bucks is pretty darn pricey. But as mentioned, cull hunts can be reasonable.
I just did a quick search and found a 20,000 acre place with 2 day cull hunts for $600 and $100 per point. If you told them you were going to do spot and stalk/still hunts with a trad bow, they may well view you as being harmless (LOL) and cut you a better deal. Can't hurt to ask.
http://www.double-d-ranch.com/whitetail.htm Maybe some of the Texas members here can help you find a reasonable ranch where you could spend a few days and maybe kill a decent management buck and a couple does.
If you can spend $2000 plus, then the options are limitless.