I purchased two new valves, one made of plastic and one made of brass. They each included a small plastic cap that is used to help remove the old unit.
I took the knob off of the control and removed the keeper clip. I then placed the plastic removal tool on the old unit and started to turn the whole assembly inside the wall. It promptly stopped and shredded the tool. The Moen faucets work by rotating a metal tube inside of another metal tube selecting some part of the hot or cold water and mixing them together to go out the faucet or shower. The tool is supposed to loosen the outer assembly so you can pull both pieces out together. Unfortunately I hadn't loosened it enough and when I was pulling on the internal shaft, it pulled clean out, without the other piece I needed to get out. Now I had a worse problem. I couldn't grab anything to pull on the piece still in the wall. I tried putting a sprinkler system removal tool in there. That got it to spin, but it wouldn't come out. I eventually stopped and thought "I wonder what I can find on google with my iphone." I simply typed "Moen Faucet removal" fortunately there was another blog post that suggested getting a 1/2" tap and tapping the interior of the remaining metal and then threading in 1/2" shaft. At that point you have something to hold on to and can use a nut to jack it out of the wall.
This is a picture of the tap and interior after I got it out:
As you can see, I bought an eye bolt and a couple of washers. This allowed me to use a large tool on the outside to rotate the whole assembly. After all of this, the insertion of the new unit was a piece of cake. It works a lot better now.
