I’m afraid I can’t help with the particulars, but I will pop in to say: Don’t delay fixing this! When I was in 4th grade, we lost almost everything to a house fire - or a house smoke, more accurately. There was very little fire damage, but lots and lots of smoke damage. It was because the switch that shuts off the furnace when it’s too hot (not the wall thermostat, but some switch inside the actual furnace) didn’t flip, and so it just kept on going until a fire broke out. Thank goodness that was one of the less than 20 days ever that my mother was substituting at my school. Instead of walking home that day, I waited for a ride from her, so I wasn’t home alone when that happened.
No matter how expensive it is (and IIRC, it wasn’t an expensive part), it’s cheaper than replacing everything you own and being out of your house for two months while they clean and replace your walls and floors.