On my Monaco coach the inverter is on all of the time, shore power or not. Shore power just feeds through the inverter as part of the design. You do not have to worry about switching it on or off since it is all done automatically. There is a manual disconnect for the inverter but there really is no need to shut it down since it also charges your batteries when connected to shore power.
In any case, the OP said that all 120 AC items are operating as they should. The only things that are not operating are those that need 12 volts DC. This involves any thermostats, 12 volt lighting, etc.
The battery readings sound correct for fully charged 6 volt batteries so the problem is either with a bad 12 volt shut off switch or with the relay that is powered by the on-off switch.
