Sounds like a bad thermostat to me.
The 110 vac part of that water heater is totally separate from the 12vdc/propane section. On the back of the heater, you will find a switch, the main 110vac line, a thermostat and a high temp limit sensor. The last two are physically connected to the tank itself.
This is the backside of the heater. Mine has electronic ignition instead of manual pilot, but otherwise should be identical to yours.
In the center of the heater, there is an aluminum box. You can see the white 110vac wire entering from above. On the top side of that box, just to the left of where the blue wire connector is, you will find a rocker switch that turns the heater off and on. At least there is one on mine. In the lower right corner of the box you can see a hol and a red button. That is the high temp reset switch. If the heater goes over temp, that trips like a circuit breaker and has to be pressed to reset.
If you remove the rectangular cover and trace the wiring, you will find the thermostat.
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