Hi
Ok, first some terminology:
A converter / charger takes shore power and turns it into
12V power to do things like charging your batteries.
An inverter takes
12V from the batteries and turns it into 120V to run thing like hair dryers and TV sets.
You may also have a generator in your MH that creates 120V from fuel.
With a few exceptions, most things these days run on
12V. The lighting in the vehicle is a good example. A fantastic fan (if you have one) is another good example. Do these things only work on shore power? The microwave is an "exception" it will only operate on shore power or off a generator.
You should have two 12V systems in the vehicle. One starts and runs the engine to move you down the road. The other runs the gear you use when parked. Normally the part that runs the engine gets called the "chassis" system ( as in chassis battery). The part that runs the camping side of things gets called the "house" system (as in house battery).
Since there are two 12V systems, each with it's own battery, it's quite possible for one battery to run flat while the other is charged enough to be useful. If the vehicle is stored for a few weeks without power, this is a likely thing.
The most basic way to troubleshoot electrical systems is with a multimeter. These days, you can get an adequate one for less than $20. You can get a really good one for a bit more. Adequate is plenty in this case. First thing to measure are the battery voltages. Depending on what you find, there are a lot of branches to the troubleshooting tree.
Bob