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Replacement parts for MSD2556AE models | AppliancePartsPros.com

I think they are referring to the drain and drip pan in the freezer floor.
I do not think this is your problem.
That drain is there so water from the defrost cycle drains down into the drip pan under the unit and then it evaporates.
You said that the problem is when the ice maker is on so I am assuming everything is OK if the ice maker is off.
It sounds more like the ice maker is overflowing.

Here is a service manual for your unit.
http://appliancejunk.com/forums/index.php?action=downloads;sa=view;down=40

Make sure that the ice maker is level. If not water could overflow at one side.

There is an adjustment to change the amount of water that enters the ice maker, see the manual.

Could be that the water valve is not closing quickly and is causing an overflow. The valve will have to be replaced to fix this.

Could be that the valve is leaking. I am not sure if you are shutting the water off when you shut the ice maker off. If you are then this is a possibility.
Just take a look at where the water enters the ice maker to see if it is dripping.