Did you check your circuit breaker? Sounds like a low battery since lights are 12V and maybe furnace is operating off 12V also if it isn’t getting AC current.
The lights and furnace are both 12v. Having it plugged in only charges the battery. The battery could be getting bad and isn’t holding a charge or the inverter is going bad and not charging the battery. When mine is plugged in I get 13.6-13.7 volts at the battery.
Do yourself a favor and order a diesel heater. Best investment you will make. My diesel heater is now my primary source of heat. Your windows will not freeze up and will not have condensation on them
you likely have a stuck furnace fan. power should not come directly from the battery when plugged in. your converter should handle the power requirements of the furnace or AC with a dead battery.