Thanks, e30pq. I checked the lock nuts on the bottom of the stabilizers, and sure enough they were loose. I tightened them up, and now the stabilizers don't move side-to-side anymore. I'm only worried that I'll tighten those nuts too much and rip the threaded rods out of whatever they are threaded into.

Winds are over 20mph today, so I'll have to let you know if it fixed the tail wag another day. =)