So since the 6990 might not be ready for full duty mining yet, should I swap my 5970 (currently in my gaming rig) with the 6990, and then plug in the 5970 and 5870 into my mining rig?
If you're using the 6990 by itself, it will work fine for mining. I'd leave the 5970 in your gaming rig (and do bitcoin mining in the background) since it won't lock up your system while mining if you watch a YouTube video, etc., like the 6990 will.
Wow what a stupid bug to have. Does this happen with every Catalyst drivers package of only with certain ones? Also have you tried disabling GPU acceleration in flash? I heard that can cause the video card drivers to lock up. Also update flash to the latest version.