Do the tank heaters have thermostats? Do they work like heat tapes and can you leave them on when hooked to shore power? Thanks for any answers and Happy Holidays!

I don't know for sure about the PC but a class A instructions I had said "continuous on  as long as there was water in the tank(s)".  The A also had elbow heaters all the way to the valves and I did not leave them on, preferring to dump at the end of stay vs leaving valves open and elbow heater on.  I did get in trouble with frozen valves once and had to stand down until the elbow heaters warmed fluids enough to allow valves to open and then dumping.