The only people who can diagnose is really your health care professional - it's outside our remit to actually diagnose what your condition is. The best we can do is suggest what we'd do. To be honest, first off I'd go see a physical therapist with a sports background, get a diagnosis, just to be sure that you haven't got a serious condition developing.
...but let's say you don't have the cash right now for a consult. First off I'd switch to dumbbell pressing & find a groove that doesn't cause pain. Next up I'd start up doing some work on the shoulder stabilisers - stuff like L-flyes, internal rotation, push-up plus (sometimes called scapula push-up). I would suspect my scapula isn't tracking properly & that's causing some pain issues in the shoulder due to compensation patterns. I've also found stuff like doing chest expander training actually seems to help scapula function (I've no idea why, I actually first read about it a VERY old book by Thomas Inch written in the early 20th century if I remember rightly, but it does seem to help), you can use bands as a substitute chest expander if one isn't available. I'd also work on things like planks & side planks - now I know you're thinking these are core exercises, not shoulder, but think about it holding in those positions work the shoulder stabilisers as much, if not more than the core & if you focus on good position it works the little shoulder muscles hard if your there for a couple of minutes with very little chance of injury.
I'd also consider doing an FMS (function movement screen) shoulder test, push-up test, quadruped & be especially aware of any asymmetries left & right, working with dumbbells should also allow you to begin balancing up any imbalances you've come across.
Basic stuff you shouldn't forget are foam rolling (especially around the scapula) using both a foam roller & tennis ball/lacrosse ball), stretching, a decent dynamic warm-up before pressing. You may want to video yourself pressing, check for form issues, we've got people like Pat Reeves onboard here who's an International Powerlifting judge, who can really pick apart your form if she can see it.
Hopefully that has at least given you some ideas, but ideally, as I said, go see someone who works with athletes & does physical therapy as they will be able to really see what's working & what's not!