I installed a new water filter first to eliminate that as the source of the problem. The problem persisted, so I ordered and installed both a new water filter head and a new dual water inlet valve. Everything works fine now. I suspect it may only have been the water filter head that was bad, but the fridge is 10+ years old and I figured I might as well do both components to be sure, and that way I wouldn't have to worry about the valve going bad later. The online videos were very helpful.