Does your fridge look like this?
Here is the PC board
Here is the 120 volt heating element where I am reading the resistance.
Have to figure out what actual model you have. What is it?
You are describing 2 problems.
The 120 VAC short is most likely from the heating element shorted. The 120 only works that element. If you have a VOM meter and know how to use it, measure the element to case ground for a short. If a short = replace.
The igniter is on 12VDC. It could be the PC board or the actual igniter coil itself.
Since you said you actually have it running on gas with match start… Then the PC is working to some level and so is the electrode itself. If you have the moldel that looks like mine, the electrode does 2 functions. One to spark, the next to give feed back the flame is lit. You seem to be missing the spark as the gas valve turned on once the match heated up the saftey circuit.
Question: When you select to run on LP only and you turn the fridge on, do you hear click, click click outside?
If you hear clicks and no light that means one thing, but no clicks at all suggests the physical igniter or the igniter is not being told to fire.
Tell me your model number so I can look up a wiring diagram and find out what you have. .
If you are not electrically friendly, do not trouble shoot this, there is shock hazards. 10,000 volts jumps an inch and the igniter has a kick to it.
Hope this helps
John
PS I see you have an 05 SD. Have to compare brake controller notes once you get your fridge fixed. Does yours lock the brakes with the manual lever?