Does it go completely off or just get dimmer? It's supposed to dim when you turn on your lights, but it sounds like it's dimming too much. Sounds to me like a defective radio. That dimming wire accepts 12V and is supposed to tell the radio it's time to dim. If there was a spurious ground you would probably pop a fuse, so it doesn't sounds like a ground issue to me.
[This message has been edited by jscott1 (edited 03-27-2014).]