Is the water heater shutting off after a reasonable time? If so, it thinks it has hot water in the tank.
Just to make sure it's clear, "turned off" means both taps are shut, not just that there's no water flowing out the showerhead. The problems come from the showerhead being turned off but the hot and cold taps left on. Some kitchen faucets might have a conceptually similar setup (though that's comparatively rare, and basically unheard of with the inexpensive fixtures most RVs use).
Maybe you have a water heater bypass valve somewhere you aren't aware of? I would be a bit surprised if the RV did not come with a bypass, though they do sometimes do most anything to save a buck or two or three.
