Fan in back not turning, front divider very hot

What you are going to find is this... Your COILS under the Plastic are Condensating and FREEZING and then this is what is causing the On /Off of the Temp. You have to DEFROST the COILS (Under the Plastic) and remove ALL the ice and Moisture. Once you get this problem resolved, it will work perfectly. This problem was caused by the Condenser Coils being Dirty and and the Switch on Top, Under the Plastic from being unable to work properly. This is a time consuming issue but easy to do. DO NOT SCRAP or BREAK the ICE after you take the Plastic off, use a hair dryer or Hot Water and make sure the Drain Tube is completely cleaned...

this is thermo switch controled change thermo switch

That is a very bad sign, but before you unplug it pull the machine out from the wall and check to see if it feels hot back there as well. Then remove the lower cover and find the condersor fan motor. Its the only fan back there. I'd willing to bet that the fan is not running. The purpose of the the fan is draw air across the compressor and blow it through the condensor therefore lowering the temperature on both parts. When its not working/ running, the heat from the compressor and the condensor is pass onward to the machine and the machine then acts as a heat sink.

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You either have obstructed coils (lots o dust) under the fridge or the fan that is supposed to draw air across them has ceased to work. Check the coils and clean as necessary, and check the fan. Maybe you will get lucky and find something caught in it. If the fan is not working, leave the back cover off the fridge and put a small window or box fan back there until you can get the fan replaced. The fan will keep the fridge running for you in the meantime.

The fan you should replace is called the condenser fan. It's the one that blows air across the outside coils. When it fails, you'll usually notice abnormal heating of the front of the divider between freezer and fridge.

the fan on the needs to be replaced it may not be turning fast enough to help keep the lines cool.

thass as good idea, but instead justreplacethe existing fan ( which is your problem)the fan is no good, so rewiring will not help any. replace the part!

check the condenser fan motor to see that the one that you changed is running in the right direction some run clockwise some run counter clockwise
if it is the right direction check the condenser for lint buildup
and if needed clean the coil
also check the evaperator for an ice and frost build up that can ocurr when the condenser fan goes bad
normaly the buildup of ice or frost will be on the top of the evaperator coil
just use a hair dryer to remove if needed dont pull on the ice if found or use a screwdriver or knife heat only to melt

That sounds like you may have something in your condenser fan motor.. piece of bag, paper and may have stopped your fan altogther.. that would be the first place to check.. right by your compressor in the back of frig.

The light for the filter is not a problem. The thing to check is that the fan in the back is running. It should run any time the compressor is running to cool the compressor. If it is, then you need to pull the front grille or toe kick off and look under the fridge. Is there a lot of dust and dirt blocking the airflow? If so, vacuum it out. If not, pull the fridge out and check behind it. Is there a grate clogged up with dust? The reason the divider gets hot is because there is a freon line running through that part of the fridge to keep moisture from forming around the door seals. If the fridge is not cooling properly (dead fan motor or dirt blocked airflow) then the freon in this line gets very hot and all the pressures in the system rise, eventually leading to the fridge not cooling. Good Luck !