Yep, I really appreciate your help. I'm just stumped. About the only thing I haven't replaced is the preheater relay. That really shouldn't have anything to do with it. But stranger things have happened I guess.
Ok, so you are not getting voltage to the solenoid when this happens until you turn the key off and then back on again. That would almost indicate the ignition switch could be at fault. Or operation control unit.
It seems like something is heating up and kicking the solenoid out. It runs, kicks out, then I reset the key and it'll do it again. Runs for about average 30 seconds and kicks out. I suppose the new solenoid could be bad, but doesn't seem likely.