I have a 2000 Odyssey with the factory Class III hitch and a Swagman 3-bike folding rack.
I mounted the trailer light connector under the hitch and it’s not uncommon to hear it drag when pulling into a driveway with a steep pitch.
I’ve also had problems with bikes dragging in similar situations. I’m considering taking the rack to a welding shop and having them cut the vertical post and add about four to six inches to it to give the bikes more clearance. Swagman didn’t recommend that, but I think they were just lawyering it.
My kid has a Honda CRV with a Swagman rack and he’s had problems with dragging, also.
I like the Odyssey and I like the Swagman rack, but the combination has been an issue. I think it’s because the placement of Odyssey’s “magic seat” causes the hitch to be lower than on most vehicles.