I would recommend getting as many ply tires as available in your size. i used to be able to get four ply in my size, but they are no longer offered, by anybody.
Depending on the tires you have, you will find the inflation pressure is significantly higher than your truck tires. Mine are on a small trailer, with light load-1100lbs, and run at 50 psig. Inflate to maximium sidewall rating, less will sway.
Too much weight on the tongue will cause sway. Some trailer and towing packages require a form of sway bar running from the trailer tongue to the side of the truck hitch. This is a trailer sway bar, not the one on your truck.
The next possible cause, aside from excessive speed, is that you've changed towing vehicles. Short wheelbase vehicles, like a Blazer, are hell for towing. For a short distance at low speed, ok. But if you are going to tow any distance, and at highway speed, you'll want a longer wheelbase vehicle, like a full size truck.
If the tires are properly inflated, I would rule them out as causing sway. For an objective opinion peddle the rig on over to your local R/V dealer. He deals with towing issues all the time, is not in the tire business, and can discuss all factors with you. He'll also have some strong opinions on trailer tires. In fact, wandering over and looking at the biggest, most expensive fifth wheel on the lot may tell you something about what rubber the manufacturer trusts. Tell him your friend Camaraderie is looking for something in the way of a 60' chick magnet R/V, in stainless steel, that will accept his Raytheon wheel pilot. That should break the ice. (g)
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Wm. F. Buckley, Jr.