I used to have a lot of neck and between-the-shoulder blades pain, mostly from working long hours at the computer. I definitely felt it on rides as well. Then I had a car accident that made it many times worse, but also got me to a physical therapist. She said I was carrying my head wrong. Major posture problem. Keeping my spine straight (tuck in chin, lift head, pull shoulders back, relax into this position) certainly has helped! So too has building core strength (i.e. strength to maintain that straight spine while leaning forward on the bike, using stomach and lower back muscles more than arms and shoulders, thus freeing up arms more for steering etc.). I've also now invested in special prescription glasses for close work, which I hope will help me not tip my head back while staring at the computer screen (as I used to do to see it through the close focus area low down on my multi-focal lenses). Does any of this resonate with what you're experiencing?

In my experience, p.t.s often have more to offer on this type of thing than doctors. Of course, to get to a p.t., one may need to go via a doctor.