Suburban water heater SW10DE still clicking after lighting

I just purchased a used RV, and I'm having some trouble with the water heater. I turned on the gas and the water heater switch that's in the kitchen, then turned on the switch on the outside of the water heater. I could hear the click click click of the igniter, and a roar of fire, but then it just keeps clicking. The fire seems to stay lit (or at least was lit whenever I checked on it), and the outside panel gets hot (which I assume means it's heating the water as it should), but it never stops clicking.

My first thought was that the igniter/sensor might need cleaning, adjustment or replacing, but if that were the case, wouldn't the heater cut off the gas flow? So could it be the board that needs replacing? Or something else?

Any advice is greatly appreciated.