I tried Memtest. I ran 2 instances of it, each on 1000MB of RAM (I have 3 total). Both instances got to around 120% without any errors...then the system locked up and went to sleep.
So, the tests didn't find any errors, but the problem occured during the tests. What that means, I have no idea.
I spent the evening backing up files. It took all evening because the PC locked up and went to sleep about 10 times. Luckily I was able to properly shut it down when I was done; I think that's the first time it shut down normally (without me holding the power button) in a week or more.
Regardless....still at a loss of a solution, but thanks for the help so far.