the pain you're experiencing might be due to how you position yourself on the bike (independent of bike fit issues).your arms/hands should guide/stop the bike, not supporting your torso weight.. Ride so your center is in your core. It increases your power transfer btw. Doing so is easier said than done. I got in the habit of holding the bar in the center near the stem w/ both hands (which is more aero than holding the brakes or drops anyway). it keeps me from locking my elbows and leaning forward . I also have a longer, more upright stem. . The issue of shoulder/neck/hand pain has disappeared. since my posture is more central and upright, I've also noted a real reduction in upper body fatigue on long rides.

the pain might also be due to a heavy helmet. the difference may be trivial if you hold a basic and a high end helmet in your hands, but those few ounces make a real difference if you're on a 40 hr ride. I spend mucho bucks on my lightweight helmet and it's worth every penny!

I wouldn't ignore or tough out the pain btw- if technique and fit don't help get evaluated by your doctor.