thermocouple on a fridge

so, my fridge is running w/o stop. everything is frozen.
it has been kinda hinky for a while, running very cold while set on 2.

so i got a replacement for the thermostat. but it didnt include the thermocouple. i did put the old one on the new stat, but still no go.

i vaguely understand the concept of a thermocouple, but mostly what i know is that when they go bad, you replace them.
so, my immediate question, while i wait for an answer from parts town, is-
are they repairable when they fail? is it corrosion that can be cleaned up, or (i think this is what happens) the bimetal connection fails and it is dead.
is the thermocouple for a fridge the same as one for an oven/furnace?

imma hit my fave hardware store and see if they carry one. they have a ton of such parts.
but any info is welcome, even just to shore up my knowledge of how this stuff works.

i hate when you spend $100 for a part that should cost $10 max and it still isnt fixed. grrrr.