Mars 302 actually hit this one right on the head. This was exactly what was happening and he called it. You start the car up when its cold and it drives ok for a few minutes and then it breaks up under acceleration. It was the fuel pump getting stopped up with dirt. When you let the car sit, dirt falls down, drive car for a little bit, dirt regathers on pump = bad fuel pressure

When we got the fuel pump out of the tank we discovered there was no sock on it. The only thing that was keeping dirt/grime out of the pump was the tiny circular metal screen that goes in the bottom of the pump. And of course it was completely clogged with stuff. So we put the new pump in anyway just as a precaution, and after we got everything sinched back up it runs fine.

We checked the tank for the missing sock on the baffled side where the pump sits - couldnt see anything. It might have fallen off and worked its way to the drivers side of the tank. Anyway, we got a sock on it and now its fine