Hello, airways nice to have somebody wanting to move to the area.
It is so difficult isn't it, especially when you just don't know the area and when you have to consider schools.
We're one of the villages to the West and we love it. Rural but not car dependent as walkable to the next village for Dr, dentist, post office, great butcher, bakery, co op etc. Plus regular bus from there. I'm NHS and 20 min drive from Shotley Bridge hospital or Bishop, about the same to Chester Le Street and less to UHND. Further out 40 min to Darlington Memorial or Sunderland Royal.
However, you do rule out a lot of housing with the garage need, means many villages you'd be restricting yourself to the new builds which are a lot more expensive.
Schools are tricky too. Mine are all in the Catholic system by going to a feeder primary but there is little choice living out this way. Dd would have loved to go to DJ but not a cat in hell's chance. If you managed to rent as others have said around Fram, Newton Hall or Pity Me then you put yourself in a good place for schools and you can take your time to find out about other areas and move with a place secured. Housing more expensive though of course because of proximity to Durham.
Chester Le Street is a good call, housing can be cheaper and you've got good schools and transport links to both Durham and Darlington.
That said either West of Durham or CLS will make things more difficult for your teen who needs to commute to Houghall. They can get the Arriva buses for free but it will be a longer commute and often 2 different buses so something to consider.