I had that happen to me once on my old chevy 2500.
I got new tires for the truck, had been running Michelin, 245/75/16 load range e, they were just worn down or sidewall cracking, can not remember why I changed.
but the new tires, same size and load range had much softer side walls, so they rolled around, I felt very unsafe and that was just driving, not towing.
I had tire dealer take them off and order another set of Michelins, when they came in, I checked the stiffness of the tire sidewalls as I had both tires side by side, the Michelin tires had a much stiffer side wall.
they drove fine,
I tried more air, everything, but soft sidewall tires will roll around on our old trucks.