I have a GE Profile side by side refrigerator/freezer.
I started getting water dripping from beneath the ice cube tray in the freezer and it was building up ice on the shelves below. I did some research and thought that it was a faulty ice-maker so I ordered a new icemaker and installed it but I’m having the same issue 
Now, in addition to the water leaking on the freezer side, I’m getting water dripping from what appears to be the Express Chill unit on the bottom of the refrigerator side. Could these two issues be related and what do I need to do to fix the problems? I really don’t want to have to buy a new refrigerator if I can avoid it!
Thanks!
Sometimes the water fill valve will leak past, allowing water to trickle into the area around the ice maker.
Try cutting off the water supply to the ice maker for a while to see if the water/ice problem stops. If it stops, then you will most likely need a new water inlet valve to supply the ice maker with water.
I have noticed this model also gets stopped up at the drain hole where the defrost drain is located.
As far as I have seen, water valves will fail to close fully, thus allowing water to ice the freezer compartment, even had this problem on newer models.