Do trailers with coil springs and shock absorbers sway at higher speeds?

My 2001 Cobra trailer arrived from the previous owner with lead weights and dumbbells at the front to increase tongue weight. I removed them and found still plenty of tongue weight and no towing instability behind a Toyota 4Runner at speeds up to 85 mph (Arizona highway speed!) I replaced the tires last year with recommended trailer tires and have absolutely no problems except in very strong gusty side-winds, when I slow to 70 mph.

I suspect the geometry and suspension of the tow vehicle are an important part of the equation - I have always used robust, heavy four-wheel drive vehicles designed for towing and have never experienced severe instability problems.

Mike