If the burner runs (gas or electric) and the bottom of the cooling unit (reached from outside) gets hot, but no cooling inside, your cooling unit has probably failed. That's an expensive repair, but still cheaper than a new fridge.
The fridge should run whenever the interior temperature is below the thermostat threshold, as determined by the thermistor on the cooling fins. It may be that the thermistor is always indicating it is cold enough, but that is a rare failure mode. Usually it simply fails to read at all, which typically means the cooling unit should be running full bore.
You can find service manuals for your Dometic here:
Dometic refrigerators - Download Dometic Manuals
2004 American Tradition; 2014 Buick LaCRosse
Homebase in the Ocala Nat'l Forest near Ocala, FL