What I should have said was the shut off valve in the wall behind the fridge is a good set up. You may have to just change the hose along with the fridge water valve. Easy stuff.
I got the new valve, installed it. No leaks, but no water comes out of the dispenser! I doubled checked that I turned the water shut off valve at the wall back on.
The one thing I was unsure of is how I reconnected the electrical wires to the back of the valve. There are 2 pairs of wires with female attachments. I can tell where each pair goes, since the male/female attachments are different sizes. However, there's no difference or markings within each pair. They seem interchangeable within each pair. I could see the front connection in a picture I took, so I know that one is correct.
Did I kill the valve? Should I switch the back ones around? Do I have another issue? I did have water at the dispenser before switching the valve out.