I do both, sort of depends on what I am doing. I have had to push my road bike bars up to the max, so they are only about an inch lower than the top of the seat. My SUB bars are a bit higher than the top of my seat.
Might have someone watch your riding positions. Are your arms locked straight? That can have side effects on your neck. How much have your ridden the new bike? Changing from a more upright position bike to a low position road bike may take some time for your body to readjust. During and post rides, stretch out your arm/shoulders and neck to offset the strain from lifting/holding your head up.
Do you use a chiropractor? A consulation with them might be helpful. I found out I have to be careful due to an injury when I was much younger. I have some calcification in my neck vertebrae.
Good luck to you.