I have the same issue on my setup. I believe it's because a motorhome steering change coaxes a steering response from the toad which is a delayed response. Initial MH steering feels like understeer, then the toad's change produces a sensation of oversteer.. It took a bit to get used to, but I expect it now.
Dolly towing is a completely different dynamic than 4-down. The dolly has a single hinge point between the wheel platform and the dolly frame, so you are not forcing the toads front wheels to respond.
I think this response varies depending on the toad and the tow rig.
Speaking of new tires, I put new Bridgestone Duravis R500HD on our GM based Sunseeker, and they are squirmy... which contributes to the steering sensitivity when towing. You have Michelins on your Ford, so that may be like comparing apples to oranges. But based on my experience, the tire characteristics will affect towing sensitivity.
Installing a track bar/Panhard rod may help, as that would remove the side-to-side flexibility in the rear axle of the tow vehicle... It would help when not towing as well.