On shore power you appear to be getting your 12v for the house from the engine battery. Previous owner did something there I think. (Perhaps because the converter was not doing it for him and he didn't fix that.)
You don't need new batteries, just need to solve the battery charging mysteries.
Leave the WH till later, since it might be just a 12v problem to do with the first issue.
Some Class Cs have a three-position dash switch for getting engine starting power from the house batteries, while others don't and just have engine and house batts in parallel whenever the ignition is on ("running circuit") What do you have? This may be what has been altered.