Sears ice maker drips down the back of freezer

I have a sears fridge model 106.71182100. The ice maker in the fridge works fine, but over time a drip causes water to fall down the back of the fridge (inside the freezer) and builds a block of ice on the cooling coils etc. As time goes on, this causes the ice maker to overfill and quit making ice.
I recently cleaned out the ice in the back of the fridge and really couldn't see what is causing this. Water flows from a specket at the top of the fill tube into a water fill cup on the ice maker. This all seems to be set up correctly.
Thanks in advance.
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I have a similar problem with an 11 year old Amana SBS20 refrigerator/freezer. Worked fine making ice for 11 years. Now all the sudden it still makes ice fine, but overflows just a bit, and leaks into the ice bin, freezing the ice in the bin & auger rock solid eventually. Have to empty the whole thing and remove ice build up carefully from the plastic parts.
Unit is level, took out the fill line which seemed clear and had no frozen ice in it. Seems to be no way to adjust the fill amount.
Repair sites on internet all suggest no way to adjust the fill, just replace the ice maker.
Morenuf
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