Propane fridge ticking when turned off

First thing that comes to my mind about a fridge and ticking is the DSI igniter. But turned off it should not be doing that. And even if it were on, and propane off it won't keep going. It would cycle off and show the check light.

I suggest taking the cover off the outside and try to find the location of the ticking. If it is hard to determine even then, get a length of flexible plastic tube a couple feet long. Stick one end in your ear and point the other end into the back of the fridge, fishing around to find the sound. When you are close, the sound will amplify in your ear.
That's how I used to find vague strange noises, whistles etc in car engines. Works well!

Once you ID the sound source, then you can figure out why.

I'd double check that it is really OFF at the fridge. Ours if on either the gas or electric setting and the gas is off and the power options are off, will continue to "tick", trying to light.

Pull the 12v DC fuse that operates the fridge, see if it stops.

Charles