You do not have a bad converter!! Stop right there. Hang onto your wallet :)
You said the lights are brighter when plugged in, so that means your converter is ok, but your batteries are not getting any converter "juice" to charge them up, and they are very low so things are getting bad if you are not plugged in and on battery only.
the mystery now is why the battery 11v can reach the lights (fuse panel) but the converter cannot reach the battery to charge it.
the other mystery is why can't the furnace get its "12v" from the converter when you are plugged in, but instead gets its "12v" from the dying battery that is not getting charged.
New one on me--maybe others here can see what it could be.
