You will need to read the rules from AMHA. They will specify which type of cart you need, and what gaits are expected. The only breed show driving classes I have had experience with call for a two-wheeled show cart, and the Morgan/Arabian/Saddlebred, etc, are usually VERY fancy, with their heads checked up high, in fine harness.
The class description will say "carriage pleasure driving" if it means the type of driving it sounds like you want to show. At least in the Arabian rules, anyway.
Good Luck! We would LOVE to see some pictures!!