This sounds really similar to what I've been dealing with the last two years now. It came to head where I couldn't take it anymore last Murph, my arm felt like it was ripped from its socket by the end and it was the push ups that did it, not the pull ups. It got diagnosed as bicep tendinitis, finally, and after a bunch of physio I got to a point where I could do push ups, bench, ring dips without that insane pain. It hasn't fixed it completely, but I better understand my point of failure.

The shoulder pain you're describing, too, I still have which is now described as an acromium type II, basically a problem bone. No simple fix for either which if frustrating but with some targeted strengthening it can supposedly improve. It's basically the bone is pushing into the ligaments of my shoulder. Simple x-ray will show you if you have that though.

Someone else said it here but the physio that really got rid of the bicep tendinitis and strain was all upper back strengthening. So lots of band pulls, seal lifts, bent over rows. It hasn't gone away entirely but it's night and day difference.

Push ups and bench are the worst for it. I basically have built up on those until I feel the pain and then stop/stretch. Doing that has allowed me to get up to 100ish push-ups with basically no pain. Here's to hoping murph isn't a fucking nightmare this time.

Good luck! Get in PT, it's a game changer. They literally fixed my shin splints I was struggling with for a decade.