The above suggestions are what usually causes the sway problem. I worked with a trailer that we struggled to get to pull straight and it turned out to be an under spring cambered axle installed in an over spring trailer. Net result? Trailer had positive camber. If you drove behind it, it looked like Mr McGoo's car with the tops of the tires angled outward. You want zero or negative camber on a trailer. I can't think of an example when positive camber would ever be a good thing on a trailer.
A couple of suggestions not previously mentioned:
1) Check the trailer alignment. A bent spring or axle can cause the sway. You can check your alignment with a piece of string. Tie it around the base of the ball and stretch it too the inside edge of the tire. Mark the string with a Sharpie. Now do the same thing on the other side making sure to measure to the same point on both tires. If they are not the same, you're out of alignment. The question is "why"?
2) If your trailer has a cambered axle (likely), the tongue height will alter the alignment tow-in/tow-out. If the tongue is not level under load it will sway. A raised tongue with a cambered axle results in tow-out. Tow-out will cause the trailer to nibble and dart, which results in sway. Consider ordering a non-cambered axle. Cambered is better under heavy load. However, if the trailer is pulled by multiple vehicles with varying hitch height, the non-cambered axle can help. Why? Without camber, the hitch height will not change the toe-in/toe-out. It will remain at zero degrees regardless of ball height helping it to pull straight.
3) Get a Hensley Arrow. Expensive but it works.
https://www.hensleymfg.com/
4) Hoping you have electric brakes on the trailer given the load. If so, you can manually drag the trailer brakes slightly from the brake controller to correct the sway. The need to do this would be rare if the trailer alignment, hitch height and load are all correct.
I put a non-cambered axle in the problem trailer and it pulls perfectly now regardless of tow vehicle. It's a Boy Scout trailer so it's pulled by every pick-up or SUV you can name and loaded differently every trip. It pulls perfectly behind all of them regardless of hitch height now that it has a non-cambered axle in it.
Hope it helps you find your problem.