It will be very hard to find the older type thermostats. You can replace it with a more modern type non contact thermostat . Many people that have restored these chest coolers mount the new non contact thermostat on the top near the doghouse cover for the evaporator.
You will need to splice the wiring up with probably a 16 gauge wire and a few crimp type spade terminals. I would remove the back narrow evaporator line cover and use that has a neat runway for your wiring and mount it in the center of the box.
Below the doghouse cover is the evaporator which which have a fan mounted to it which circulate the cold air into the cabinet. I ended up moving up higher so the doghouse didn't cover it making it easier for and adjustment ...
Others with more experience may chime in hope this helps!!
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