Would a dead house battery disable 12V when on shore power

Some older converters require a good battery to work. Your 2011 is probably new enough that no battery is required. Batteries in the condition you described could pull the converter output way down, and possibly damage it.

You can tell if the batteries are 6 volt by checking the wiring. 6 volt batteries are wired in series, one negative to ground, it's positive to the other battery's negative, while 12v batteries are wired in parallel, positive together and negatives together.