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

Trailers are funny beasts, especially long trailers.
Shocks are important items as is using real trailer tires if your having problems; Trailer tires have stiffer sidewalls to help with the sway.
Make sure they are at max pressure.
The trailer C.G. is also very important; the C.G. must be well ahead of the axle, this where tongue weight comes into play. The tongue has to be heavy; how heavy? heavy enough to minimize the sway effect.
If you cannot picture this CG thing draw a picture of a trailer and move the CG ball aft of the axle and think about the CG being back there...When it starts moving back and forth it wants to go divergent, not good.
My Cobra trailer came with two large sandbags in the front of it and I cram all I can into the front storage space, self rigger, tow out gear etc and bungee it all down.
Flub, try and find a old school trailer man in your area, I bet he could look and your rig and have a solution.