Replaced water valve assembly. The assembly was super easy EXCEPT for the fact that my refrigerator was a 1996 model and the two outlet connections had threaded fittings on the original part, not the push-in kind like this replacement part. It was a little scary cutting those threaded fittings off and pushing in the cut tubes, but fortunately it worked without issue. I didn't need a single part from the "extra parts bag" that came with it. A little extra documentation describing the possible installation variations for older models would have been very helpful.
Nevertheless, the hollow ice cubes are gone and I'm back to full size, solid cubes that aren't sticking to each other. Water flow through the water dispenser side is noticeably stronger now as well.