You are correct in assuming that it should shut down the burner when the desired temperature has been achieved.
My first impression is a faulty TPR valve. If replacing that (one of the cheaper options) doesn't fix it, then I suspect the thermostat control module may be bad.
Is it an Atwood or Suburban? If it is an Atwood, look at the
Atwood Mobile Service Manual for assistance in troubleshooting and replacements.
I would not think it is the thermocouple as that would prevent gas from ever getting to the main burner if it could not detect heat coming from the pilot.
