Check the ice tray before using the water dispenser to see if there is ice (or the water level if not frozen) and then check it again after using the water dispenser and check if water has filled and is overflowing in the ice tray.
It may be filling when it shouldn't be.
Here's a link to another question in ifixit which has a service manual that may help. Your model uses the same GE 31-51982 mini manual which I can't find and I was hoping that using the diagnostic tests in a GFE26nnn series mini manual would be close enough to get an idea.
Fridge not cooling, freezer is really cold
Note: Check in the back of the cabinet for the correct mini manual that may be there in a plastic sleeve taped to the cabinet wall somewhere. In the past GE used to put the mini manual behind the removable kick plate (toe plate) in refrigerators (and also in dishwashers), but if there is no kick plate I've read that it is back in the compressor area somewhere, but I don't know for sure if this is correct.