I had a 68 C10 (chevy 1/2 ton) that I installed anti-sway bars and rear overloads. Even though I was still grossly overloaded the handling improved immensely and I had a lot less sway.
Campers cause our trucks to sway. How much depends on our trucks suspensions so we get them set up, or buy them already set up, to carry our campers as safely as we can. But even the best ones don't corner flat and you don't want to push them around corners. They are very top heavy wheather they wallow or not. They are more prone to roll over than cars, unloaded trucks or Suvs and we need to drive accordingly.