I've had a lot of shoulder issues in both shoulders. I've learned a lot over the past several years, especially over the past year, about what's best for me.
I dont know exactly what your issue is, but I would avoid doing heavy lifting that makes it hurt if I were you. Also I think you should stop overhead pressing for now.
If it continues and you do decide to see a doctor, I would run like hell if they suggest surgery as the best option.
There's a chance its a minor tendon/nerve/muscle issue. If that's the case and the MD diagnoses it correctly, he'll probably put you on NSAIDs and maybe muscle relaxers and tell you to stop working out. Then he'll prescribe physical therapy, which will not likely do very much.
If I had your symptoms, I'd see my chiropractor first to see what he thinks. I wouldn't just see any chiro. Mine is trained at the highest level of both ART and Graston Techinques. Its likely that he'd be able to fix it using one or both of those. I also trust him to tell me if he thought I needed an orthopedist to examine me. I dont know if this would be best for you or not, because I cant recommend one I trust unless you're in the Minneapolis area - then I would recommend my guy.
One thing I would recommend is this program:
This guy successfully treats people like professional and college level baseball players. He recommends no overhead pressing when you're having shoulder pain, and no bench pressing with a barbell (dumbell bench pressing is ok once you get back to normal).
I recently laid off barbell bench pressing for ~4 months and did some workouts from the Buchberger program, along with a lot of other shoulder rehab type work, and dumbell benching, plus several ART and Graston sessions with my chiro. When I started bench pressing again, it took me about 2 workouts to get back to the level I was at before, and my shoulders feel a lot better.