Another possiblity is where you are placing your hands and at what angle. If you are riding with the elbows bent/outward, thumbs slightly turned downward (in order to hold on to the bar with the thumbs to reach brakes with fingers) then you may be over using the rotator cuff. That will also cause neck pain in that you are recruiting other muscle groups to help give more support the neck via the upper trapizius/levetors. Also, gripping the bar more than you think over a longer period of time will cause some over-use syndrome of the smaller muscles of the arm/shoulder. The muscle on the thumb pad is a very over-used muscle and we forget to baby it/stretch it. That is where hand yoga comes in handy. I teach it in my classes about everyother week or so.

Do you have a chiropractor who can to check for misalignment?

I agree; don't wait and let it get worse. Consult a professional and get a second opinion.
Terry