Just my two cents, GE refrigerator water line freezing in the freezer door is not uncommon. It sounds like the valves are working since you say you can hear it trying to dispense. Pull the toekick off, find the water line connection, and disconnect it. Close the freezer door and turn on the dispenser to see if water squirts out the water line. Have bowl handy to catch the water. If water comes out then you have a frozen water line in the dispenser area. GE makes a heater kit to fix this. ...don't know the part no. off-hand. Rich

Hi, I pulled your parts diagram. The auger motor, which is located in the freezer door dispenser area is most likely your problem. I've replaced these in the past. Some of these refrigerator ice dispenser motors get moisture in them and fail to work properly. If you call a professional to do this you may be looking at $200-250 range. hope this helps, Rich