Thanks everyone for the great advice! I just finished draining and refilling it and it works like a charm now!

I ended up doing three things differently the second time around: I installed the o-ring on the filter (I completely missed that the first time), trimmed the pickup tube about 1/4" (I measured and it had just under 1/4" clearance already but that seemed a bit tight for such a large diameter tube so I cut it shorter), and replaced the Schaefer fluid with Audi G052162A2. I doubt that the Schaefer fluid caused any of my problems, and I'm sure it's a great fluid, but I figured since I was already re-draining the transmission and getting more of the original fluid out, it wouldn't hurt to refill with completely fresh OEM fluid.

All in all, I'm guessing the only thing that actually caused the issues I had was the missing o-ring.. But whatever it was, I'm good to go for a while now!

I've said it before, but I'll say it again: this forum is a FANTASTIC source for anyone who owns one of these cars!