My relentless pursuit to cure trailer sway - the dancing TT

I had a stock car hauler that use to do that and we determined that the wheels were toed out. A good front end shop can check it and bend the axle to correct it. As the other folks have pointed out, soft tires can be an issue, as can having too much weight behind the axle.

You might try having a buddy tow it with a different truck to see if the problem persists. That way you can isolate the problem to the tow vehicle or the trailer.