When I first started getting lower and lower with my bars and staying in the drops longer, I had this issue too. At the time, my helmet had a visor, when I removed it I could see further down the road without lifting my head so much. Pain went away.
Moving on to a recent helmet purchase, I forgot about this. First couple weeks of riding with it the sore neck and shoulders came back. It didn't have a visor to remove, but I did find that adjusting the small straps inside the helmet along with loosening the what-a-ya-ma-call-it strap that goes around the back of my head allowed it to sit on my head with the front a little higher so I again could see further down the road.
Hopefully next time, I'll remember to make certain of my forward visibility with my head down when buying my next helmet. Of course some might consider that may be exposing more of the skull to a crash, but if we are that worried, maybe we should be wearing something more on the lines of full face protection similar to a football helmet.
However for all this long winded reply, there is something to be said for just giving your body time to get used to that position and muscles getting stronger in those places. Off the bike exercises for them might speed up the process.