Although the problem ended up being the valve, it took quite a bit to finally figure that out. Because the valve was leaky, water would drip into the ice maker and freeze there on the intake pipe, eventually blocking that pipe with ice - hence the leak.
It was a simple fix once I had the valve, but one thing to note: The repair video shows the valve with push to lock connectors for the water out, and the technician in the video says to cut off the end of the tubes (which have a plastic compression fitting). I did that, then noticed that my new valve did not have push to lock fittings, it used compression fittings like my old valve.
Fortunately I was able to re-use the plastic nuts I'd cut off. I wish I'd noticed the connectors were different from the video before I cut the nuts off.
Here's the full story of how it was diagnosed. The water line to the cold water outlet has a push to lock connector in the water line just below the freezer door. It was wet, so I thought that that was the cause of the leak. No, it was only wet because the fan blew water over it. While pulling the fridge away from the wall, the copper intake pipe cracked so I had to replace that.
I eventually called Sears and they sent a technician who immediately diagnosed the problem as the valve. $129 and money well spent. He asked if I would like for them to replace it so I said yes - only fair, they diagnosed the problem, let them fix it. $522 - what? OK, go ahead. Part was ordered and a date was set. The part wasn't available so it was back ordered.
A week after they were supposed to have come back to install the valve, it was still back ordered. Cancelled that and ordered the part from an online parts place which said it was in stock. A week later, it still hadn't shipped so I cancelled that and ordered from Appliance Parts Pros who came through and delivered. The fridge was fixed that evening.