If you are having problems with lights and furnace, then it is either a converter on it's last legs or you have several bad connections. You can see some of what can happen (as in things to look for) in the picture below. Burnt connectors like this are due to corrosion which creates high resistance and reduce voltage to the loads (dim lights). These were output power plugs from the PD converter I replaced. Your PD would use simular plugs. Connections to lights, switches, and junction boxes can also have these problems. While my old converter was working, I do have 4 house batteries that I wanted to keep properly charged. For battery charging, the technology of the old linear PD converters is far behind todays modern 3 stage converter.
Side note: Your dodge chassis connectors under the dash can exhibit problems like this also.