After doing quite a bit of research on the internet and talking with my Dad, it appears that there is no way that the urgent care doctor diagnosed my shoulder correctly. It sounds like my lower deltoid may have partially detached from the bone -- not a problem with the rotator cuff.
In any case, it hasn't appeared to heal at all in the last three weeks. I'm planning on seeing an orthopedist. From the sounds of it, surgery could be required.
Our urgent care system is in need of, well, urgent care. Angela and I have each visited several urgent care facilities in the last couple of years, just to be mis-diagnosed every time.
How is it possible to be wrong every time? If you estimate that the average doctor makes a correct diagnosis 80% of the time, then the odds of these urgent cares failing us 5 times in a row would be: .2%^5 = .03% -- very improbably. In other words, either urgent care doctors are significantly below average or the urgent care system prevents them from succeeding (or both).