I would not replace a cooling unit in a 30 year old refer.
Thermostat is in the refer compartment. When it gets to the set temperature it will shut down either the flame or turn off the electric element.
There was many recalls on Norcolds. Did they ever get done?
You could replace cooling unit and that may not be problem. Older refers have their cooling liquid solidify and will not start again. I had an old unit that I took out and rolled on the ground. It worked for a while then completely died.
I would get a new direct replacement unit. It will be much more efficient. To me it doesn't matter whether it is a Norcold or Dometic replacement.