I would try a hardwire for the stop solenoid right to the battery for a little while. I haven't looked to see if this theory is plausible, but I wonder if it could be a flakey connection to the stop solenoid from the "start" position of the ignition switch that is a good connection on the "run" position. If the connection in the "run" position was flakey, the engine would cut out while driving.Andrew