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.