The American Driving Society, has no affiliated clubs in your area. That doesn't mean there aren't any though.
On facebook there are several driving sites, you must join, but it is worth it.
Draft Horse Sale, Trade, Adopt, Stud, and Tack
The Carting Zone
Carriage Driving Online
Carriage driving classifieds
Ask if anyone knows of some clubs or driving instructors anywhere in your area, you will also find out about how much carts and harnesses cost. Don't be discouraged though, most of the items for sale are the really good stuff that is hard to sell locally, you can find more cost friendly tack locally on Craigs List.
Good Luck, just getting to be around horses, is a step in the right direction, plus it is really good for you.