Late follow-up ...

Rode again last night with someone (my LBS manager) who has previously observed this issue. We both realized that when I'm in the drops, my shoulders are level. I just naturally bend my left elbow to the same degree as my right elbow and everything is fine. I prefer riding that way so there's no issue most of the time. But when I'm on the hoods, the right shoulder droop is noticeable.

He correctly observed that my right elbow is in a good bend but that I have a tendency to want to lock my left arm straight thus forcing my left shoulder up an inch or two. His concern is that when I do that I also am probably "torqueing" (his word) my torso, i.e. twisting it very slightly to the right. In doing that I'm probably also twisting my pelvis to the right which has repercussions in the way I sit on the saddle, the stresses on my legs and knees, etc., etc.

He's in the bike bidness to make a living but didn't recommend a fitting and liked the taping idea. He suggested giving it a try and/or using some Kinesio tape.