Well, two suggestions actually.
One, the whole icemaking unit can be replaced fairly easily. If you are at all handy, you can probably do it yourself. Just buy a new unit from a parts store. Make sure to check the water control valve for proper function.
Two, (and this is the one I recommend), buy some ice trays and do it the old fashion way. I work in the water supply industry in a quality assurance role. One of the most prevalent causes of bad tasting water is icemakers. The tiny supply line and intermittent flow thru it combine to degrade the taste AND quality of the water going to the icemaker. Most refrigerators now have small carbon filters on them to attempt to offset this degradation. Problem is, once the water goes thru the filter any disinfectant (chlorine) in the water has now been removed. You no longer have protection from bacterial contamination.
While I was typing this, I did a google search for 'Kenmore side by side'. WOW, icemaker problems seem rampant.