You might also want to look at the springs and make sure one of them is not broken! I have some issues with an equipment trailer, and an RV travel trailer that both had broken springs.
Tires will also cause sway, especially if one of the tires has a broken inner cord. ST trailer tires are also not as good as an LT - light truck tire!
Unfortunetly with 15" rims, LT tires with wt carrying capacity do not exist as much as 16" LT tires. I in fact, got rid of my 225-75-15 trailer tires on my equipment trailer, and went with a 215-85-16 10 ply/LR E and the trailer rides better empty than I have ever felt it ride. Especially empty!
A hwy style rib tire will handle the best typically if you go with an LT tire. Otherwise, Dico builds good trailer tire, I have Coopers on three of my trailers, Dico on the fourth!