Shower – Identifying old single handle shower valve/cartride

I've done exactly this, including stripping the valve stem. You can remove it!

Most valves require a special wrench to unscrew. This is a cheap single-purpose tool; you probably won't be able to use an adjustable wrench or general purpose socket. Here's an example.

You'll slip this wrench over the valve stem and onto the valve body, then unscrew using a screwdriver stuck through the other end of the wrench. Make sure you have a snug fit with the correct size; there are two sizes in common use in the US and most valve wrenches have both.

As an alternative, it is possible there's a retaining nut holding in the valve. If this is the case, there would be a small ring just on the inside of the outer valve body. It would have notches to allow you to unscrew it using snap ring pliers (though you might be able to manage with a pair of needle nose pliers held at an angle). If this is the case, you should be able to see and move the locking ring. On the other hand, if you see a hex outline when you shine a flashlight down past the valve stem, you'll need the valve wrench.

Remove the seats, put pipe joint compound or Teflon tape on the threads, and reinstall. If it still leaks check threads inside the rough-in-valve (housing in wall) that the seat screws into. If those threads are bad you have 2 options: You can use 5-minute epoxy on the threads or you can replace the faucet.