The full story:
Dishwasher stopped running mid-cycle. When I first began troubleshooting I tried manually advancing the timer knob, and the machine would run. That led me to believe that the actual timer assembly was bad, so I replaced it. The dishwasher ran fine for a day, and then the same problem came back. This time, however, I also had to push in on the door to get it to run. I looked around online, and someone suggested that the door latch wasn't closing tightly enough. I adjusted the door latch tension (that little latch plate on the inside of the roof of the dishwasher), and tested it again. It worked, but only for one wash. The dishwasher refused to run after that, unless I pushed against the door firmly while it was in its cycle. I took the door apart a third time, and when I checked the door switches, one of the buttons was collapsed. Since it was busted, I pried the switch open, and found that the spring for the contact plate in the switch had melted into the plastic for the button piece itself. I ordered two new switches to replace the ones in the dishwasher. Once they arrived, I installed them without issue, adjusted the door tension to reduce how hard the switches were being depressed, and then it was off to the races. She's run without a problem ever since. On a side note, I tested the original timer and found out that it still works. Now I have one spare door switch, a spare timer, and a working dishwasher, all for $120 and a few days of hand-washing. I also didn't have to shell out $500 for a new machine. In my book, that's a win!