Water is trickling down through where the ice dispenser depressor sticks out of the freezer door. The ice is getting soft. Water is trickling down under the framing for the ice /water dispenser on the freezer door.

Wait til dark and take a flashlight put it in the ice shute and close door and look up at ice door if you see light then warm air is getting in, if no light turn off water at wall, if it quits leaking replace waer valve. check door gasket on freezer especially at bottom for tears

By not being there seeing the amount or flow rate i'm not sure but from where your saying it's coming out, It sounds like this is either condensation from behind the framing or the flap that closes when your not getting ice is not closing. This could be a broken spring but if the flap is closing. There is a heater blanket behind the framing that heats and dries the condensation out. If it goes out, water will leak out from where your describing. Try going to www.partselect.com and enter your model number. This should bring up diagrams of your unit. See if there is a heater that goes in your model. if there is, I'm pretty sure it will be bad and need replacing.