[clutch492]

I know you said that it has fuel pressure on the fuel rail, but I had this same problem and there will, and was, always be a little fuel pressure on the fuel rail. However, the fuel pump was not running. Here is the cure for the problem you are having.......

If it is the same as the 96 bonneville, there is a ground connection bridge under the carpet, on the drivers' side, right near the front of the door frame. pull the carpet out from under the plastic trim located beside the seat. You will find a connector with about 8 to 10 ground wires. take off the black cover. you will find a strip of metal that connects all the grounds together. pull it out. You will see corrosion on some of the terminals. clean them with a wire wheel and reassemble. These are all the major ground connections for the car. The fuel pump grounds here as well. This is what is causing your electrical problem and why your car is stalling. I discovered this while searching this site in the past. And, it was correct. It cured the troubles with the bonneville I was working on.

There is another bridge (like the one on the drivers side) located on the passenger side. Check all of these grounds if you want, but the fuel pump is grounded on the drivers' side.