You probably have some type of rotator cuff injury refering pain to your hand. There are some tests you can do at home to confirm this like trying to lift your arm up away from your body which will hurt you, or making a 90degree angle at the elbow and then pressing the outside of your fist against a fixed object. O/w it could be a brachial plexus injury which is doubtful
I'm not the doctor, but to me it sounds like you need to see your doctor. Your symptoms appear to indicate that swelling or other issues in your shoulder are aggravating the nerve that runs down your arm into your fingers. Are you doing anything to treat the shoulder injury like taking advil and applying ice periodically? Are you able to keep the arm and shoulder relatively motionless as with a sling to keep it from getting reinjured? How did the injury happen? The reason I ask is because if it was something violent like a car accident, the problem may truly reside in your neck rather than your shoulder.
Just an FYI - chronic pain docs generally treat pain that has already been diagnosed by another doctor and is of at least 6 months duration and not expected to improve in the foreseeable future. Check through the list of Med Help forums and look for an orthopedic or family medicine forum for more specific advice on your injury.