Sway is an entirely different problem from towing capacity. It's possible to sway your way off the road sideways with a 1500 lb trailer that is bad, yet drag straight as an arrow a 6000 lb trailer that is well designed and set up properly.
That being said, I'm going to guess that your first problem is that the truck's rear tires are under-inflated. This rig, at a sticker weight of 5,828 lbs, behind a TV rated according to the sticker on the door at 6,000 lbs, is right at the limit. So I think you have too much trailer for the truck.
But beyond that, if things were set up right, your main problem would be accelerating, stopping, and going uphill, not swaying.
Assuming you have the right tires on the TV, you can't load it up like that and NOT pump up those tires, especially the rear ones, to much higher than you run with the truck empty. If you failed to do that, the fix is obviously really cheap. At least as a first step.
