If one has an issue with trailer sway, it rarely ha anything to do with the tow rig rating. 99+% if the time, it is and has something to do with the trailer. IE too little hitch weight, tongue too low, so trailer us not level to name two most common issues.
I've towed a 12k trailer with a truck that had a 12500 gvwr, no sway, no white knuckles, frosted over 20k lbs, even had my four kids ages 5-12 with me! Another 2000 or so lbs in the bed.
Another time towing the same trailer, total weight 8500, with an 11800 empty MDT, 26k gvwr, 35k gvwr! Trailer was all over the effing place! Went across scale, truck weighed 11800 with trailer attached, trailer axles 8500. I had no blinked hitch weight! Moved bobcat 2' forward on trailer, no sway, no problems towing the trailer.
Another issue not talked about, tires that are too soft in sidewall. NOT just P metric mind you... I got talked into putting four Michelin XRV 14 y tires on rear of MDT. Sidewalks were soft so motor homes had a cushy ride! Hauling loads and trailers also was horribly effected.
So, as long as new trailer allows you to be under axle weight ratings, trailer is setup correctly, even if over the gcwr/tow rating, you should not have sway or handling issues. Will you go slower up grades vs a light setup, everything else the same. Of course you will. Change from a with.in rating 3000 to 6000 lb trailer, you'll notice that difference too! Just as changing out to a taller trailer with more frontal area that weighs the same. Do you all know, it takes the same amount of increased HP at 60 mph, loading your rig up an additional 1000 lbs, vs adding 3 square ft of frontal area!
Lots of reasons one may be having a bad tow experience. Being over, at, or under tow rating, means squat!
Marty