It appears the converter is is working to supply that 13.7 to the monitor somehow, but that the fridge and furnace are on battery.
Something is blown so the converter's 13.7v is not reaching the fridge and furnace and also note the battery is getting lower with use, as indicated by the voltage drop when the furnace comes on.
You can keep going that way until the battery bank gets too low and then you will need a battery charger, or else get the converter back on line.
You don't need new batteries now, just a battery charger.
BTW the fridge on 120v does use a little 12v for its controls