Neck Pain

[BleeckerSt_Girl]

Changing the seat position (up/down) wouldn't affect my neck, would it?

You should not be raising or lowering your saddle in order to help neck pain.
Your saddle needs to be at the right height to make for correct leg extension. Any deviations from that can cause other problems such as knee pain (too low) or rocking hips (too high) while pedaling. There is very little "play" available in seat height, anything too off the mark will cause discomfort.

Raising or lowering your handlebars *might* help neck pain, as can adjusting your reach by various means. And yes, handlebar width and hood placement can affect your neck and shoulders.

Have you given your body a chance to ease into the new fit before rejecting it? Sometimes any little change in body posture can make us feel sore at first, until our bodies adjust to the new position. A new weight distribution or body position will ask various muscles to change a bit to adapt. Often new positions seem odd and uncomfortable at first.
Perhaps you could give it a few more rides to get a better feel for it? If nothing else it will help clarify further what you like and don't like before going back again to the fitter.

Or maybe you can even ask for a different fitter... a second opinion, like people do with doctors?