I have a similar GE fridge myself and it had the same kind of problem a while back. I found a service information sheet tucked under a panel at the top of the fridge that told me how to read the trouble codes from the refrigerator and that led me to a failed fan that blows cold air to the upper part of the cabinet. Another symptom I had was that I had ice forming in the bottom of the refrigerated section (not the freezer part) due to the lack of air circulation.
Note that there are multiple fans in that appliance and the ones that cool the freezer are different from the ones that circulate air in the refrigerated section, which would explain why your freezer and ice maker are working fine.
Replacement fans from GE were really expensive, so I pulled the dead fan and checked for manufacturing information and found out it's the same fan used in many computer CPU coolers. It was an order of magnitude cheaper to buy that exact same fan from a computer supply house than it was from GE.
So look under any covers that are easily accessible and see if you can find a service sheet; if not, you should be able to search online for information on your specific model on how to read the diagnostic codes and interpret them. That should point you in the right direction; specifically you should look for the fan speed readings and check for any that aren't getting data or are reading zero for the speed.