Mine does sort of the same thing- intermittently though. It reads zero when the key is off but starts to work again as soon as I turn the key clockwise. Sorry sarcasm aside.
Beyond a lose connection it sounds like a bad motor in the instrument cluster. If it's under warranty they will replace the whole cluster. If it's out of warranty you can either try and replace the whole cluster or just the little motor.
Their was a long period of time that this was a known issue in Chevy trucks. Their are a few shops that just do Chevy instrument clusters. They have the motors and typically have a 2 day turn around. I'm betting a call to one of those shops would provide some better insight if this isn't under warranty.