On a small trailer like your 1550, the exact type of tires probably won't have that much effect on sway. But if they are ready for replacement, you definately want to get ST's. There've already been several threads on this subject in the Towing and Tow Vehicles section. I encourage you to look back and read them.
But in your case, I suspect the cause is a bit simpler and the fix is free. You didn't state what year your T-1550 is, but I looked in 1993's brochure here in the "Files" section to get the GVWR of a 1550. If your model year is different, your GVWR may be a bit different, but the rules are the same and the calculations are the same, and the fix is the same.
You also didn't tell us if you are using a weight distributing hitch or not, and if you already have some kind of anti-sway device on your towing setup.
The most common cause of sway is not enough tongue weight in comparison to the GVWR of the trailer. Because of this, your trailer can be too heavy behind the axle. This can cause sway when you hit a bump or encounter wind conditions. In short, the tail wags the dog.
It's compounded by a short wheel base tow vehicle like your Jeep, but that usually can be fixed by adjusting tongue weight, especially in lighter weight setups like yours.
Tongue weight should be at least 12% of the GVWR, and preferably around 15% to be sure. Much more than that is going to cause a different set of problems. The '93 T-1550 is spec'ed at a GVWR of 2085 pounds. 15% of that is about 312 pounds. Your tongue weight should be right around that. If it is not, the fix is free and easy. Shift some gear around in the trailer to increase the weight on the tongue.
It is also helpful to determine the actual loaded weight of your trailer, as you may have way too much gear in it, but for starters, get the tongue weight up to around 300# or so. If that doesn't stop the sway, then you have a different set of issues to deal with.
If you have a WD hitch, you will compensate for that by taking up another link or two on the chains if the setup sags at the hitch because of the heavier tongue weight. If you don't have a WD hitch, and you get sag at the hitch point, it's probably time to consider upgrading to a WD system.
If adjusting the tongue weight doesn't fix the problem, then it is time to look at an anti-sway device, but the tongue weight is most important, and needs to be fixed first.
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