I have a 4 x 6 utility trailer. with the trailer full to the top, it pulls great. have an empty trailer or just half full and it sways back and forth. It has pulled smoothly on my car for many years so I knew it wasn't the trailer. It is the aerodynamics of the bed and trailer bed causing it. If a new cap on the truck doesn't fix it, I will have to redesign the trailer. Maybe you have a similar issue. Do you use a
tire rack? Can you adjust the position of the car to change tongue weight?
And get the tires balanced to eliminate that. Are you sure the tires were new?