When the ice maker fills with water, some water also runs out the water spout in the door.
Locate the dual water valve (shown on the following picture) on the back of the refrigerator. Disconnect the wires from the brown solenoid and watch the water dispenser for a few hours.
Post the results.
Here are the breakdown diagrams and Replacement parts for MAYTAG MSD2555DRW REF - SXS/I&W | AppliancePartsPros.com
Gene.