Bathroom Tune-up

Something in our cold water tub faucet assembly wore out and now the faucet is spinning 360 degrees. It still works, but you never know the best place to stop when shutting off the water. What is even more odd, is that it seems that the hot water is going much, much faster; after about 10 minutes the water cools off considerably.

We have a two wall-mount faucets and a stopper in the spout. To remove the faucet I simple turned the entire assembly counterclockwise. Since the left (hot) faucet works fine, I haven't touched it. When removed the situation looks like this:

Now, I am left looking at a closeup of what I believe to be the stem. In the past, I have removed this and replaced the washer or O-ring in the back, but that was always to replace a faucet that couldn't stop the flow of water. This works, it just doesn't stop spinning. We have been using the tub in this configuration for two days since we can turn off the water by turning the cartridge by hand. (I know water might leak back there, but I want the motivation to get this done.)

I thought it might be a problem with the handle assembly, so I removed the functional handle and its associated cartridge stopped at the correct place without the handle. The inside of the handle looks like this:

I am sure I need to replace some component of the non-functional cartridge, but I am not sure how to even determine the make of my handle. I can't find any clues to who makes this except for four numbers on the escutcheon. Desiring any help that anyone can provide.

Thanks in advance. On another note I did replace all the caulk around the tub, which helped out nicely. It is always good to have at least one satisfying job to counter a frustrating one.