Number 93 in the diagram below is the part where you select your options on the dispensor. You can pry this off from the bottom. There are no screws holding this piece on. Most of the time you can grab it with your fingertips and pull up on it and it will come off.
You are going to find you problem is with the solenoid that opens the flap. More than likely rust has developed on the shaft of the plunger that goes into the solenoid.
This prevents the flap from completely closing .
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Click this link and check page number 25 to see freezer door disassembly procedure:---
