My mom had a shoulder injury - by the time she went to a doctor, it was almost at the “frozen shoulder” level of non-movement. It sucks, but you need to get it checked.
I love doctors. My dad was a doctor. That said, a doctor will probably do jackshit for you in this instance other than prescribe pain meds and refer you to a physiotherapist. If it’s really bad and/or you have good medical coverage they may order some imaging, which is often not terribly useful for soft tissue damage.
Immobilization is useful sometimes but you can’t say “thats how you fix a shoulder injury” because there’s a hundred different types of shoulder injuries. You haven’t even told us the exact location of the pain and the activities you describe that cause the pain don’t really isolate what specific movement is doing it.
Might be the AC joint. I had similar symptoms. Fixed with simple surgery to remove excess bone. As almost everyone else has said, see a doc.
If you don’t have a lot of spare money or good health coverage definitely cut out the middle man and go to a physiotherapist.