You have pain in your shoulder....could be something, could be nothing.
IMO I think your best best is to see a physical therapist if you live in a state that has direct access and allows you to see one without the referral of a physician.
The PT can do a comprehensive evaluation and can determine and refer out to a MD if they think imaging is necessary (MRI, X-ray, etc).
They can palpate the soft tissue and bones, check for imbalances and restrictions which may contribute to your pain.
And they can treat you right then and there, show you some exercises/stretches that may be able to reduce your symptoms. They can watch you while you perform activities and can make adjustments/modifications if necessary.
I'm not sure your primary physician can do all that. Referral out to an orthopod or referral out to a PT!. Pain meds which will just mask the problem but you've already said you don't want pills.
If you go to the orthopedic MD, first off, good luck getting an appointment in the next few days since you are a non-emergency....sure, the ortho can also do an evaluation...maybe show you a exercise or stretch or two, but then what? If he's conservative, he'll do what....refer you to PT! Complain enough and he might send you for some tests....
So now you have to wait again for an appt for imaging tests.
And then wait for the results....then wait for the doc to get you in to discuss your results....if it's minor then hey let's try PT first.....sigh.
Best bang for your buck....see a PT....you can still also make an appt for the ortho if you want, but you'll probably be able to get a few sessions in before your MD appt.
Either way....get it checked out!