There is a heater kit "bandaid" fix available from GE to dress this issue...and yes it is freezing in the door, most likely right behind the cradle in the small recessed area.

you can use a hairdryer to warm this area and thaw it or I use a zoom spout oil bottle filled with hot water...the tube off the bottle is the perfect size to push down the dispenser tube and flush out the ice plug...reg will be along directly to link you the fix kit, I am sure...

No Water from the Dispenser on a GE Refrigerator

Source: Hotpoint SxS fridge, M# HSS22IFPACC