So I installed a new video card (nvidia geforce 9500 gt) and it worked great! -- except with the problem described by the title, namely that I was seeing the entire system freeze from time to time when playing Flash video from youtube in full screen. I did not see it happen with non-fullscreened video and the video quality didn't seem to have any effect (although I did not test exhaustively).
Interestingly, I had been having problems with my previous card, so I had already turned off hardware acceleration. I really didn't want to hang on to a Windows machine just for watching Youtube videos, so I tried a few other small things to no effect. To make a long story short, I eventually turned hardware acceleration back on and it hasn't crashed since.
I was kind of surprised by this result, since everything I had thought up to this point indicated that hardware acceleration was more of a liability than an asset, so I thought I'd record my experience just in case someone else was experiencing a similar problem.