Yes #35 is the defrost thermostat.

Here is your tech sheet.
https://www.servicematters.com/d...0Sheet%20-%202255774.pdf

Run the diagnostics on the tech sheet.

Since the problem started as an intermittent I would suspect the control board or the defrost thermostat. but there are other parts that will do this
Sometimes you can actually see that the defrost thermostat is bad, it will be bulged out.
A defrost problem usually does not give the early symptoms you describe as once the coils are iced up they will not defrost if a component is totally shot so the fridge will not fix itself.

You have to do more diagnosis to narrow down the cause.
Remove the evaporator covers and take a look at the coils.
If they are heavily frosted/iced up you probably have a defrost problem.
If they are clear and it still is not cooling the problem is elsewhere.

Is the compressor and it's fan running?
If yes are the evaporator cols evenly frosted up?
If they just frost up partially especially if it is where the freon enters the coil you may have a low freon condition.

Is the evaporator fan running? If not this may be your only problem.
Check with a meter to see if it is getting voltage.
If yes, then the fan is shot
If no, then probably the main control board.

Here are a couple good sites with basic fridge repair.
http://www.applianceaid.com/refrigerator.html
Refrigerator Repair Guide: How To Fix a Refrigerator - ACME HOW TO.com

Does this look like a control box problem ?
No as it sound like the unit is trying to run or may be running.

Or could it be a freon issue ?
Could be that all the freon has leaked out.
If it was just low freon you usually will see part of the evaporator coils get cold/frosty. This could explain why the cooling efficiency dropped (as the freon leaked out) and now has finally stopped all together

Is there a service manual for this refrigerator ?
I cannot find one perhaps someone else with a source will jump in.
Thanks !
Jim

I did not see any major bulging on the defrost thermostat.
None of the evaporator coils are frosted up
Evaporator Fan is running
It does not sound like a defrost problem as you do not have an iced up evaporator.

Neither side is cooling ( Refrigerator / Freezer )
Only the freezer side actually cools and then the cold air from the freezer is used to cool the fresh food section.

Check that the compressor windings have continuity.

Thank you for the replies ! The compressor was hot,
I did not see the part youve mentioned, AP3083556
but this part (AP3885081) rattles a lot when I shake it.
What do you think ?
Thanks !