I have neck issues also. A football injury from over 40 years ago suddenly started hurting. Exercises given to my by a physical therapist help a lot. Still I decided to get another bike as the handlebars on my Trek 5200 was a little low even when I jacked them up some.
I got a Giant Defy Advanced. I also looked at the Specialized Roubaix and a few other relaxed geometry bikes. Even the Giant wasn't quite right so I went to a more angled stem that raised the handlebars about 1.5 inches more. Now I'm good.
Have you tried flipping your stem? That cheap fix should raise your handlebars about an inch alone. If that doesn't work, look into a taller stem.
Warning: don't go nuts with jacking up the handlebars. I put on a stem extender on my old Trek 5200 that raised the handlebars about 3 inches. It messed up the geometry so much at it induced a high speed wobble. That's when I decided to buy a more relaxed geometry bike.