If you have 12V power when connected to shore power, that has to come from the converter, if the battery is not supplying it. That would explain things if all of the 12V worked from shore power. Since the refrigerator doesn't but everything else does, that would make me believe that you have a separate problem with the refrigerator. If your fresh water pump, lights, and the furnace all work from shore power it would seem that way. I would trouble shoot only one problem at a time.
A wet cell battery is considered to be fully discharged when the voltage reaches 10.5V. To accurately measure the battery voltage, one of the battery posts should have all wires removed and then by measuring between them you will get accurate battery voltage. Have you checked the electrolyte level in all of the cells? If they are low that will reduce the amount the battery can hold, even when charged to 13+V.
Did you test the jack to see if it will work from the battery, or from the converter/shore power?