With the BAL light trailer leveler under one wheel and a decent amount of pressure from the stabilizers, our high wall is pretty stable most of the time. It does seem to vary and most of the time (I have not kept records) it is the unpaved surfaces that make it so.
I made some stabilizer platforms that keep them from having to be cranked so far down after doing an over/under axle conversion that added ~5" of height to the frame. I made them 8" so the stabilizers have to come down not so far. I don't know that this helps so much but I didn't have them with our first PUP and remember it bouncing quite a bit, even with a good bit of pressure on the stabilizers. Your frame can play a huge part in it as well.
I would try just a bit more pressure on the stabilizers and highly recommend the BAL leveler, not just for ease of use but for helping to knock down any wheel movement.
Platforms:
BAL leveler:
It is my opinion that trying to stabilize the bunk ends directly will not yield you much. If what they are attached to is still wiggling, you're really not going to do much for the bunk ends when they are anchored to the box still bouncing.