Those little black things next to your stater solenoid are relays... I had one go out on my 87 ford bronco II ... fuel relay and it would also run for about that long then die. Very frustrateing. I think the relay cost like $30+ back then and replacing it fixed the problem. Just in park and letting it run it would die... but the relay would get hot before it died and clicked a few times....
I am sure that you have fixed the problem by now... but I thought that I would add my two cents anyways.