Should batteries be disconnected when on shore power?

You need to fix-repair-replace the inverter charger.
Or if the inverter is still working you could just get a converter to charge the battery and supply 12v power.

If this seems like a bit much you are best to just haul it in for repair.
Again if the batteries are sitting at 10.6 volts they need a charge asap. Get a 10+ amp portable charger if needed to get them charged up. When charged they should sit at 12.65 volts.