Ok not to rain on anyones parade about the MN Hooved animal rescue- but I would never adopt from them. A lady I know adopted 2 horses from them, and when she moved them from the pasture on one side of the road to the other side of the road they confiscated the horses because she "failed to notify them of the horses being moved" to another location. You never own the horses, they always belong to MNHAR and you cannot make any medical decisions for them, even end of life decisions in an emergency. They can pull the horses anytime they feel like it.
There are other horse rescues in the state, but there isn't very many draft horses that come into rescue.
One place you could check out would be Great Expectations Ranch in Pillager. Great Expectations Ranch
They buy and sell auction horses and usually get a good feel for them. I've never seen them sell a horse for more than 500-600$.