I have some experience in this arena going from a 5000 lb trailer (in-laws) to a 8400 lb trailer with a "half" ton. Short answer: No, a heavier trailer does not help with the push-pull from semis for my "half" ton. Cross winds have quite an effect on my rig, though my experience with those are in Eastern Oregon and the Columbia River Gorge. They push the big rigs around too.
That said, I'm not swaying. Just getting pushed a little. Hitch is an EAZ Lift 1400 lb bars and 2 friction sway bars, though I think my lack of sway is more due to the heavy tongue weight of a toy hauler.