Late 80s early 90s GE refrigerator
Yes, it sounds like you need a new thermostat. They’re easy to change.
The model number is on the sticker on the left wall of the refrigerator. You could see it in your picture number five.
The thermostat may very well be unavailable from GE, but you’ll probably be able to find one online from a generic source.
These were fairly reliable, basic refrigerators. They do tend to dry out food a lot worse than new ones and they use about twice as much electricity, but they get the job done if you’re not too picky.
John