I tow 3 different trailers. 17ft Ice Castle fish house, weighs about 4500lbs, 25ft travel trailer about 6000 lbs, and an aluminum 12 ft trailer for side by side. The travel trailer I use a weight distribution hitch with sway bars, it still does not help. The 17ft trailer is the worst, almost scary at times. The 12ft trailer tows fine. I towed all 3 of these trailers with my Tundra using the BF Goodrich K02 load range E tires. No sway at all.
That is why I was surprised when I bought the trail boss that it tows so badly. I figured it must be the load range C tires.