Well, more work went into the laptop yesterday. I have gotten it to the point where it seems to be working. I further localized the problem to the graphics card on the logic board, which means if I really want to fix it, it will cost me $400, not worth it. Basically, if I push on the graphics card, it goes.
When I took it apart, I kept trying things such as disconnecting the internal screen and other stuff. I also oiled one of the hinges. Then I started putting it together, piece by piece, testing it at every stage. I eventually got it completely back together and could not get it to fail. This could be a good sign. I only need to to last at most 8 more months. Here is to hoping it will.