I don't think the run /kill solinoid is bad only because I have never seen one do that and it would flash a code , I think that the voltage which makes it hold in is not releasing when the ignition is turned off , first guess would be diode in the alternator went bad and it is getting a voltage feed from the alternator , or the relay is sticking , another thing that I think that can make that happen is the RPM sensor as if it is bad it would tell the controller that the engine is not running , but if this were the case it should flash a code or two and if you have deluxe instrementation your tach would work , let us know when you get it right as I don't remember seeing this senerio on the board before -----------