Trying to fix a leaky shower. How can I remove this cartridge?

My experience level is novice at best. I've done basic stuff: installing faucets and toilets in my own home while remodeling. If it's just a matter of taking things out and putting new things in, I can handle that. Never messed with hooking up drains or water lines, I leave that to the professionals. That said, this seems like an "easy" fix (haha) if I can just get this cartridge out and put a new one in.
Symptoms with my shower:.
When this valve is turned to "shower", a significant amount of water still comes out of the bathtub spout.
Water from the spout is pure hot water.
Diminished showerhead flow & lack of hot water.
What I've tried.
Turn off water to the house (I have no access to these lines in the wall, there's no access hatch).
Unscrew the hex nut on the shower/tub handle, and removed the handle.
Unscrew the bits holding the outer plate in place and remove all that.
Photos below are what I'm left with. This is embedded fairly deeply into the wall and I can't see any further in. I don't know if there's a clamp or another hex nut holding things in place and even if there is, I'm not sure I could get to it.
It doesn't seem to want to turn counter-clockwise anymore. I didn't force it because I don't want to break anything.
Basically, looking for advice before I proceed. Thanks in advance!.
Little more info: This house was built in Michigan 1977 and this is the original fixture. I don't see any sort of brand, serial number, or other identifiable markings on it at all.
Question from user AWT at stackexchange
Answer:
Your next step is to get an adjustable wrench and turn that larger nut counterclockwise and remove the valve stem. Then you can turn the the valve clockwise and remove it out the bottom of the valve stem. The valve you have is similar to the one shown below from Home Depot and is included so you have an idea what your dealing with. The black packing you see in your picture will have to be replaced as will some washers and "O" rings.
Answer from user JACK at stackexchange

My experience level is novice at best. I've done basic stuff: installing faucets and toilets in my own home while remodeling. If it's just a matter of taking things out and putting new things in, I can handle that. Never messed with hooking up drains or water lines, I leave that to the professionals. That said, this seems like an "easy" fix (haha) if I can just get this cartridge out and put a new one in.
Symptoms with my shower:.
When this valve is turned to "shower", a significant amount of water still comes out of the bathtub spout.
Water from the spout is pure hot water.
Diminished showerhead flow & lack of hot water.
What I've tried.
Turn off water to the house (I have no access to these lines in the wall, there's no access hatch).
Unscrew the hex nut on the shower/tub handle, and removed the handle.
Unscrew the bits holding the outer plate in place and remove all that.
Photos below are what I'm left with. This is embedded fairly deeply into the wall and I can't see any further in. I don't know if there's a clamp or another hex nut holding things in place and even if there is, I'm not sure I could get to it.
It doesn't seem to want to turn counter-clockwise anymore. I didn't force it because I don't want to break anything.
Basically, looking for advice before I proceed. Thanks in advance!.
Little more info: This house was built in Michigan 1977 and this is the original fixture. I don't see any sort of brand, serial number, or other identifiable markings on it at all.
Question from user AWT at stackexchange
Answer:
Your next step is to get an adjustable wrench and turn that larger nut counterclockwise and remove the valve stem. Then you can turn the the valve clockwise and remove it out the bottom of the valve stem. The valve you have is similar to the one shown below from Home Depot and is included so you have an idea what your dealing with. The black packing you see in your picture will have to be replaced as will some washers and "O" rings.
Answer from user JACK at stackexchange

