Wall furnace assit please

Hello all. My issue is with my wall furnace. it is a single wall gas furnace, one of the tall ones. It recently started turning it's self off at random or not kicking on when I turned the thermostat back up. So I pulled the cover off and the wiring seemed very brittle and so I replaced that with new wire. (the heater is older) While doing this I noticed that the Blocked Flue Sw. 350 Deg (at least that is what the name of the piece seems to be when I googled a pdf of the parts) twist around inside the oval metal piece that attatches to the furnace it's self. I noticed that it almost seems to be shorting out, if you manipulate it just right it will run for a while but then some times it will shut off again and need to be wiggled. Does this just mean that part is shot and I need to order a new one? Or is something else going on here. If I can fix this problem for a small cost and NOT buy a new furnace that would be WONDERFUL!