Howdy from the high desert,
It would be VERY helpful to know make and model of the appliance over the stats of the trailer, the refrigerator
if a three way uses a different tsats per class of circuit, ie. a 12 tsat for the 12v and a 110v tsat for a 110v too...if its a two way 12v/propane or 110/propane it is different.
If the tsat is closed (on call of cooling) then disconnect all power and turn off the gas and test for continuity, a twelve volt (battery powered "self contained power") light
indicator style auto tester will work or a volt/ohm/continuity meter...if you get continuity or the light
glows then move on to the wires leading into the pilot box, there is a heater in there usually 200 watts to 400 watts depending on your size of cooling system and voltage of heater element/ elements, its a round silver cigar about 3 to 5 inches long and poped into a neat tube soldered directly to the right spot on the refrigeration system, test the two wires coming out, if burned out it will 98% of the time show no light
or infinite resistance (open) "you must disconntect both wires to test" You then need to replace it. There will be two of these in the case of a dual voltage unit, one for each voltage. I have found that old cracked wiring at the terminal block can be a simpler fix and should be looked at first.
Next... on some models you cannot run both gas and electric because there is a MECHANICAL interlock system excluding or denying one system from running when on the other system.
..On other models there is a manual toggle switch for operator switchover...If yours is a toggle, test the toggle switch and is connections
...If this has a cam/tab sytem be sure what position is the cam/tab...some really old ones use a microswitch against the cam...Alot of the time I am patient with new owners of old systems because the guys who built them did not use the same play book so individual attempts of manufactures to be crafty confuses new owners...the cowboy manufacturing process in the old days ment variations...really look at this area.