Sounds like an issue for a person when I first started posting on here 10 something years ago. The main posters said add a sway bar. I said to add hitch wt. I got chewed out for not recommending a sway bar, only doing the hitch wt. Person added hitch wt, and could go to 70 mph with NO issues. He then added a smaller sway bar kit for added measure.
I've found with the single axel trailers I used for my landscape biz with mowers etc, that getting into the 15-18% range vs 11-14 or so for a dual axel, that it helped keep the sway at bay.
Some other issues NOT mentioned. How is the side to side load? if one side is heavier by say 400-500 lbs, then that could contribute to sway also. Also possible. the tire portion of the axel is not in alignment. rare but possible. Along with one tire being having the tread shifted, other is running smooth, one is lumpy, very slight, but still able to move a trailer sideways.
Marty