I did 3 things at once, some combination of which appears to have stabalized this system (knock on wood, I guess): downgraded to Testing, installed Debian's generic i386 kernel instead of the Athlon optimized one, and disabled the second NIC in the system.
Generally when troubleshooting you want to minimize the number of factors that change at any given step of the process so that you can isolate the failure point. However, given the potentially long timeframe between me taking an action and being able to verify whether it had the desired result, and given that I really need the system to be working, I just took the "shotgun" approach - eliminate as many failure points as possible in an effort to restore minimal functionality.