If you are only draining the freshwater tank, the rest of your trailer's water system will retain a "full charge" of water in the lines. The water pump is a "one way" device and draining the water out of the freshwater tank (using its drain valve) will not draw water "backwards" out of the trailer's water lines, including the hot water tank.
For safety purposes, I don't recommend leaving the electric element "on"...in fact I don't even use mine. The water heater uses very little propane and it only takes half a second of current in the electric heating element to burn it up if there isn't any water in the water heater's tank. The switch for the element is a PITA to get to on most heaters (outside, behind the access panel and usually behind a pipe and a bunch of wires) and I don't even bother with it. Your LP heater on the tank shouldn't be run on an empty tank either, but, at least the control switch for that is on the inside panel and should be off if you don't need hot water.
I normally only turn the LP burner on once a day for about 15 minutes. I turn it on at night after we eat and by the time we've gathered the dishes, the water is hot and we turn it off and wash the dishes. It will stay plenty hot enough for the rest of the night, unless we are going to take a shower...if a shower is on the menu, we turn it on for the duration of the shower, then off again. Those little tanks are very efficient and extremely well insulated. The water is still pretty warm in the morning for what we need it for.