I believe I have eliminated most of the magnetic and PSU coupling, but the oscillations are still there. I grounded C4, the compensation capacitor connected to the transformer, and the oscillations decreased dramatically. I have reason to believe that there is some sort of coupling through the transformer, so I will buy a better transformer type with my next Mouser order and see if that fixes it. Thanks to anyone who gave suggestions, you guys taught me a lot of stuff about eliminating noise and coupling!
