Being honest I think the compressor is getting power because it is hot and if it is getting too hot to hold your hand on then it is getting enough power to run. It sounds to me like the motor inside has locked up and that is a very expensive repair. The fan motor and the compressor are on different circuits and both connect to the main board. I do not believe replacing the board will solve the compressor problem because the compressor runs on 120 volts ac, the fan is dc. If you ordered from repair clinic you can connect the board and try, they will take it back minus the shipping even if it is connected.
What I expect you will find is the fan will run but the compressor will not. I think it will need to be replaced to make the repair. You will need to call around because prices for the labor for that job can vary greatly. Just ask 2 questions.
Do you have an EPA issued license to recover coolant?
What is the minimum you charge to replace a compressor?
They will need the model number, I looked up the kit number from GE and it is WR87X10211 part number and I have it linked below. Sears was the cheapest well known company I could find for the part. $250 plus shipping.http://www.searspartsdirect.com/partsdirect/part-model/GE-Parts/Refrigerator-Parts/Model-PFS22SBSBSS/0432/0161000/G0805029/00002?blt=06&prst=0&shdMod=PFS22SBSBSS
I would expect you to find the labor for that job to fall between $500 to $700. There is one part that may get it running for a little while and that is called by a couple of names. 3n1 or hard start. I can tell you that though this part may get it going it will not last long, it is not designed to work with the type of setup you have and it will not last more than a couple of months. I do not recommend letting anyone install this part. The reason I bring it up is because a lot of techs still use these. They were fine for the older refrigerators of 20 years ago when refrigerators were not energy effecient. Not todays, so do not let someone do that. What that will do if it does happen to start is cause the coils inside the motor to burn, then charred bits will travel through the cooling system, then your refrigerator will truly be finished.
Please ask if you have any questions,