Shower valve won’t stay shut off

I have an old Moen shower valve (not sure of the exact model, but it appears to be several decades old) that won't stay shut off. If you push in the knob to shut off the water, it opens itself back up within a few seconds.

Some online sleuthing suggests that taking out and cleaning or replacing the cartridge might fix the problem, but I'm unable to figure out how to remove the cartridge (maybe there is no cartridge). Here's a photo of what it looks like with the knob removed:

There is a clip (visible in the photo above) that holds the valve stem in place, but even with that removed I still can't figure out how to disassemble the valve any further. Nothing seems to move if I put a wrench on it, and I can't wiggle anything out of place.

Any tips on how to proceed with the disassembly, and/or other ideas on how to get the valve to shut off properly again? Should I give up and just replace the whole valve?