First I would go to the game and fish dept of some different states you are interested in. For instance, Texas Parks and wildlife dept:
http://www.tpwd.state.tx.us/ . You should be able to find the answer to most of your questions. But they should also have a "contact" link on the website where you can send them specific quesions. Most are good about answering specific questions.
Texas is basically private land hunting, but there are a lot of animals you could hunt. From whitetail and turkeys, to feral hogs and many exotics (axis deer, sika deer, blackbuck antelope, elk,and many kinds of african game). Many ranches have free ranging game, but many are high fenced operations. If you want a specific type of hunt, you need to be clear with prospective outfitters what you want.
I also do swap hunts, where I take people hunting here on my ranch in exchange for them taking me hunting at their location. You probably won't be able to do that, but I know people swap other things than hunts. For instance, you might swap lodging in England for someone who wants to visit your country. That way they could save big hotel bills.
Good luck!