I have been noticing some soreness and swelling in my elbow. It feels stiff when I wake up and after I exercise. Would you suggest I should see a rheumatologist to understand my condition better?
I suspect you have Olecranon Bursitis
This can be managed in primary care
Yes.
Yes. Not to be too trite, but after discussing with your primary care provider, a visit to rheumatologist would be appropriate. Rheumatologists specialize in joint problems.