I've got the trailer wiggle a few times and it's always turned out to be not enough tongue weight.
... agreed. There are devices like below to help control sway, but i'd HIGHLY suggest you try either try moving weight foward or adding weight near the tongue. For light trailers I like to have at least 15% on the ball/hitch.
Another thing to consider is the hitch height - a properly loaded rig has the side rails of the trailer perfectly horizontal with the correct % tongue weight and when the trailer is attached to the tow rig
. Think of the trailer as a seesaw with the wheels as the fulcrum. If the nose is even just a bit higher than level it is taking away tongue weight.
Leveling the trailer is accomplished with a shank that either raises or lowers the ball to the proper height. They sell shanks with a multitued of drop/rises usually in 1" increments or you can get an adjustable one like this. This one (as most) can be flipped 180* if a rise is needed instead of a drop. A big carpenters level laid on the side rails of the trailer gives you a much better measurment of "level" than eyeballing it.
Also, because the lever on the tongue side is so large compared to behind the wheels, once the trailer is level, Adding a small amount of weight near the tongue can make a big difference. Lead diving wieghts can easily be bolted to the underside ...
... I once towed a wagon back from Chicago to NY on a rented Uhaul trailer. There was no way for me to move the wagon any more foward and the trailer started wagging at 47-50mph. We stopped at the next exit and purchased 4x40lb bags of rock salt and ratchet-strapped them as far foward on the tonge as possible. We were then able to safely travel at 60-65mph which over a 1,000 mile trip made a huge difference not just in time on the road but relaxation level while behind the wheel.