So I took the old machine to Altex this morning for a consultation. They said it would cost me $120 for parts & labor for them to do it, so one of them suggested a cooling bed, which I had never heard of, but it cost me only $20. It sits under the laptop with two movable fans so you just blow air on the outside of the computer to keep it cool--you position the fans wherever you want depending on where your computer's hot spots are. Since I always just use it for a desktop anyway, it was a good solution. I've already run some very intensive stuff (malware & adware scans with Malwarebytes and Spybot S&D) and it survived without shutting itself down due to overheating. I'll have to get used to the new positioning of the computer, but it looks like I'll be able to keep this one going for a while yet. Also now that the old fan is completely dead, it's much quieter. I can't even hear the cooling bed running, and it has two fans on it.
I also had them look up exactly what kind of RAM I can put into this one for an upgrade (DDR333 SODIMM), but I didn't do that yet.