No, it's never on when tank is empty. We camp all summer and the water heater isn't emptied till winterizing. It is a Suburban. Had four heating elements and two thermostats, so far. Works fine on propane. My mechanic suggested just using it on propane, but that does not sit well with me, since it is a two way heater.
My mechanic is the one who suggested this problem is caused by the heater getting too hot. Seems the t stat should stop that from happening. Also had it repaired by the dealer. They know less than the mechanic. This is the 2012.