2006 Chevy Cobalt engine does not turn off automatically

Well after looking at the wiring diagram for you car it looks as if you may have a body control module (BCM) problem. The only way you are going to find out is to do some tests with a digital multi meter next time it happens. Now the body control module is in the center of the dash below the radio. It has a brown wire going in from the ignition switch and a brown wire going out to the power control module (PCM) which is under the hood near the radiator overflow container. So what you will need to do is first check to see if the brown wire from the ignition switch has power or not if it does then ignition switch is the problem. If not then check the brown wire going to the pcm if it has power then the bcm never told the pcm to shut off so the bcm is the problem. Then you need to check the white wire at the bcm as this is the off/crank/run signal wire. If it is keeping power then the bcm is not telling the car to shut off. Let me know how it goes