All converters (not alternator) are not equal, so hard to say how normal yours is. If you have not heard the converter fan running often, in the previous several years of ownership, then you may have an issue.
If you have a digital volt meter, what does it read at the battery posts, with the converter in use. A reading of 13.2 is a float charge, and anything over 13.6 is indicating the converter is charging to catch up.
You say your camper is at a place with available shore power, so if plugged in, not disconnect switch shut off, your battery should be at full charge, each time you return.
If you are not plugged in while away for a month, yes your battery is depleted, may be damaged.
Jerry