Curious to know what did you do to determine there is no current from the harness when the engine is started? Do you mean increased voltage? You can't measure current unless you have the solenoid hooked up and it is actually actuating. On mine the voltage on the one wire went from 0.04 to 0.07V and the other went from 11.6 to 14.4V when I started the vehicle. Those are both relative to frame rail ground.

Sorry, I do not know what the duty cycle is of this solenoid and at what conditions it will be open or closed.

I think the stalling is another issue completely. Do you have any other codes?