My battery has died a couple times over the last couple of months.

The radio will be powered without turning the car on provided you have not gotten out of the car since shutting it off. Once you open the driver's door, the power to the audio unit should cease. If this isn't happening, then yes - there could be some kind of a draw occurring. Keep in mind that you should never turn your radio on without the engine running, or use any other accessories, as the battery will lose power precipitously. Certainly, see your Chevrolet dealership for assistance if you continue to have concerns.

Does the ‘dome light’ illuminate each time the drivers door is opened?