Just some ideas:
1-Unless you’re working with DV footage that may see further layering and compositing, I would try working with QT reference movies exported from Avid. Save some space on your hard drive.
2-If your final footage doesn’t have to carry an alpha channel, I would try exporting using Avid’s codec from AE. I’ve had issues with Avid coded ALpha channels, so I use Animation codec for alpha stuff.
3-Double check that all video footage (even ref movs) imported into AE have proper field interpretation (i.e., lower field first for NTSC DV and SD footage).
4-You probably know this, but be careful of thin lines (less than 4 px) and ultra-sharp (as in vector art) graphics. Check the motion blur box when animating imported photos or vector art. Check the field blending box when slo-mo’ing imported video. Apply a little blur to sharp-edged vector graphics.
5-Avoid multiple renders of footage when possible.
6-Try doing more compositing inside the Avid. Yep, it’s painful sometimes, but the right mix of AE and Avid can be beautiful.
I’ve seen a few posts like yours here and on Avid lately. I’ve used AE with Avid for a short 3 years, but haven’t encountered the issues I read about from other posters.
Best wishes