Tire pressures all around definitely affect sway. Trailer tires, truck rear tires, truck front tires, all must be inflated enough to carry the load. If any pair is under inflated, you will get "directional instability."
Probably the most important are the truck's rear tires. Second most important are trailer tires.
Pickup trucks empty are light in the back. Fully loaded, they are heavy in the back, and the tire pressures have to be adjusted to compensate. If you are attempting to tow a heavy trailer with the same rear tire pressures you run normally when empty, you all but guarantee sway.
The importance of tire pressures was brought home to me years ago when some friends and I went out to get some firewood in his 15 passenger van. We loaded up about half a van full, and set out for home. I was driving, and I needed every inch of the two lane highway, it was wandering so badly. I asked the owner what pressures he was running, and he didn't know. In the years he'd owned the van, he had NEVER checked the tire pressures.
We stopped at the first gas station I saw, and I discovered his tires were all in the low 20s. They were all way overloaded even when the van was empty, let alone loaded with firewood. Pumped 'em all up where they should have been, and the rest of the way home was perfectly stable. The difference was impressive to say the least.