GE Monogram icemaker not dispensing

Now another issue has cropped up with my 14 year old GE Monogram built-in side-by-side fridge.

The icemaker bucket is not dispensing ice through the door. I removed the bucket last night and allowed any build-up to melt. Re-installed this morning, with ice in the bucket, and when I press on the dispenser, I can hear the motor making noise, but nothing comes out. I am assuming that something is "frozen" between the motor and the shaft that drives the screw in the bucket. Any ideas?

The GE website says to hold the switch for a full minute, but seems like that might harm the motor.