Initially, the fill tube (3/4" diameter white tube in the freezer) was filling with ice. Now we are getting "puddles" of ice around the icemaker or the icemaker is over filled with ice. I defrost it (by removing the icemaker and/or fill tube, and it works fine for 3 to 4 days, then starts freezing again.

I believe that I need to replace the icemaker solenoid in the back of the refrigerator, left side near the floor. Your parts diagram shows two solenoids:
AP4009555 kit, water valve, and
AP4013716 Valve, primary water

1) do you agree that this is the likely problem, and,
2) which one is the part I need

Thanks