My rear wheel shifts in the dropouts, what could cause this?
Check the alignment of dropouts, and if QR mates with dropout with full surface on both sides. Maybe you could use marker or some kind of fresh ink, paint the dropout, install everything and then observe if paint transfer was uniform.

If you have spare wheel (or you can borrow one) ideally without QR try installing it and see if problem persists. If the problem persists, than the issue is on the frame side, or in the way you are tightening. Otherwise, it is on the QR side.

Lastly, I have to ask this, are you sure you are installing and tightening QR right way? (finger tighten, and then use lever on QR to get a good tight)