I am thinking from your description that the coil is bad on the solenoid.
The pictures really help. Otherwise it would be good to know what model year your RV is, and brand. If before 1995, then it is likely that you have a 'converter' that disconnects the coach battery from the lights while plugged in, but this looks a little more modern, with a battery charger that is electronically voltage regulated, and much better.
This relay probably is energized to turn it on or off, otherwise is not using any power. Check with your manufacture, or the place you bought it for a replacement relay. Expect it to cost $50 - $125. It is pretty special. Most solenoids consume power while they are on, and could drain the battery pretty quick.
Good luck,
Fred.
