New graphics card installed!  It was a pretty messy installation, as I expected.  A few system crashes later, I seem to have it all sorted without any issues.  Updated drivers, and now for the testing phase.  I just played Oblivion for a good, straight 15 minutes without any crashes.  That's not enough to say that I fixed it yet, but I'm going to try out Bioshock Infinite.  I REALLY REALLY REALLY hope it doesn't crash.  If it doesn't, that means two things:
1.  My old card was indeed dying and this new one fixed the problem.
2.  I can finish playing these games!!!

Let's just hope it's all going to work out.

Edit:  IT WORKED!!!!!!!!!!!  I am so relieved.  I just played over an hour of BI straight without any crashes.  I think my frame rate dropped just a tiny bit, but it's still very very smooth.  DDR5 memory has more "bandwidth" than DDR3, I learned, so even if it's 2gb versus 1, the performance isn't as strong as the DDR5.  Oh well.  It didn't matter much, because with my old card, I couldn't play anything either way.  I'm so happy.

I had a really crappy day at work, but I feel a lot better now that I fixed this problem.