Of course drafts can gallop - just it takes a big man up there to sit it out. Going round corners is the tricky bit.
Draft horses are descendents of the chargers used by Knights of Old - all that Agincourt history stuff. Then in later centuries they were used for farm work. In the UK many are still used for work up amongst the forests.
If you seek a draft horse then look up the Internet for the UK horse magazines - draft horses are unlikely to die out as a breed.
Suffolk Punches, Clydesdales, Shires, Percherons, Irish Drafts, Friesians - they are all to be found. But they don't come cheap (by US standards).
When you find what you want, then ship it over to the US - it is possible.
But whatever you do, make sure you've got a set of strong boots - those big fellas feel pretty heavy when they stand on your foot.