OP update: Man, this thing is difficult to diagnose. I basically found the problem though: The problem is the fuel pump relay sometimes does not energize when you turn the key on, and therefore the engine is not priming. So, I replaced the relay. Still not working. I check all the wiring to the relay, I even bypassed some of the factory wiring, cut the factory connectors off and spliced on a new plug for the relay. Still not working. After some research, I found out that the relay is grounded through the computer. Basically, one side of the coil is connected to the positive battery terminal at all times. When the computer grounds out the other side, the relay is energized. When the computer does not ground out the other side, the relay is de-energized. I ran a new wire from the relay to the computer. STILL not working. arrgh! So, I replaced the whole computer and....yep, STILL not working!!
Then I remembered I've had problems with relays in this truck before. Could it be some OTHER relay is the problem? In the fuse box, right next to the fuel pump relay, is the ECU relay. Hmm, if the ECU relay is bad, the ECU won't turn on. If the ECU won't turn on, it won't ground out the fuel pump relay. If it won't ground out the fuel pump relay...I'm going to get another relay tonight and replace the ECU relay. If that doesn't fix it, nothing will. While I'm at it, I'm replacing ALL relays in that box! This truck is getting dangerously close to getting sold. If I can't get this problem figured out, it's gonna have to go. My mom was stranded fo 2 hours yesterday before my uncle came and picked her up. Even worse, she's borrowing my cars now.