I used to get a lot of popping in my right shoulder.
This is most likely due to a kyphotic, shoulders rolled forward posture. At least it was in my case.
It went away when I started doing a lot of backwork, but this is what I think you should do.
Start with this warm up everyday. Save the pec stretch for after benching, but do everything else.
Second, continue on with pressing, but I’d prefer you stop… but it won’t kill you, just make your recovery slower.
Press 2 days a week, 1 bench 1 OHP.
On bench days do the warm up, then do another warm up of 2 sets of 25 light machine (or cable) rows. Then bench. That’s all the pressing you get to do right now, the rest is back work. Start with 5 sets of 10 machine rows, as heavy as you can, but get 50 reps. Follow this up with 5 sets of 20 facepulls, and 5 sets of 20 band pull aparts with the THUMBS UP. the band pull aparts are very important. do some light external rotations either with a dumbell or cable and you’re done.
OHP - warm up, then lat pull downs, 2x25, then press, once again, that’s all the pressing you get to do. Then follow this up with 50 total chin ups. As many sets as it takes to get there. Use a pronated (hands facing each other) grip. This should decrease shoulder wear. Feel free to substitute with pull downs if you want. but go 5x10 on those. follow this up with 100 reps of facepulls, and 100 reps of band pull aparts THUMBS UP, and some light external rotations.
On off days, or lower body days, as soon as you walk in the gym, do the upper body warm up, and 100 band pull aparts. Off days, just do it at home.
I should also note that your hips are almost definitely tight as well. Check out my low back thread, and the same way you get over a creaky low back, is the same way you restore proper hip posture and eliminate pelvic tilt.
This will fix you, I’m serious. Give it 3 months, although I think it will likely be good in 2 months, but work at it everyday, and your shoulders will be fine and your posture will be great. Your upper back will be so strong you’ll likely hit new PRs on bench, and you’ll have the shoulder flexibility to OHP with ease.
Straight on till morning.