It sounds like you have a bad "12V Battery Isolation Solenoid". It is the thing that makes the loud "click" when turning the house power on and off. Whether clicking or not, something inside might have failed and it is stuck in the "on" position. They generally look something like this.
There are 3 & 4 terminal versions available but 3 is more common for your application. One terminal is for power coming from the house battery, another terminal is for the power going to the house. The 3rd terminal is the wire to the control switch. See if you can locate yours, then find one on-line just like it. Disconnect your house battery before taking off any terminals.
For trouble shooting purposes, I suggest tapping hard on the solenoid with a small hammer but do NOT dent it. The "jolt" might just free it up internally. If it works again, you know for sure it is your problem.
If you are good with electronics, you can measure the voltage at the output terminal using a volt meter. A primitive method would be using a 12V light bulb to ground. The input terminal from the battery will always have power. The output terminal going to the house should not have power when the control switch is turned off.
I had a solenoid fail in our first motor home. Something shorted inside it and it got so hot that it charred the firewall it was mounted to, and also melted the wires around it. My point here is, if it's not working right, you really need to replace it for safety sake. You don't want to just forget about it because your RV runs the risk of catching fire. That is unless you disconnect the battery that supplies power to it.