I think that will definitely answer a few questions. Personally I would hook the batteries up and then with shore power un-plugged turn on something with high power draw like the furnace for a minute or two. This will remove the surface charge from the batteries.
Turn off the furnace, wait a minute or two and then take a voltage reading across the battery terminals. Make note of that reading and then plug in shore power and take another voltage reading. Hopefully your reading with it plugged in is at least a full volt higher than unplugged.
