Just to jump in here...

I once had a Dodge Omni that did this same thing. Being pressed for time I took it to a old guy that rebuilt alternators, generators and starters in his shop. These used to be called Generator shops and there might be one near you.

Anyway, he started it, let it die, and said I needed a new grounding strap. This was a flat naked wire that ran from the block to the body. He took mine out and it was burned almost in two.

The new strap fixed it.

I asked him what caused this and he just said it would take all day to explain it and if he was fixing or talking he charged $35 an hour, so I still don't know why this took place.

All I know is I never had the problem again.