In reply to DaveHK:
https://www.campsites.co.uk/search/campsites-in-northumberland/newcastle-up...
This might be the closest to being in between the 2. Nearest pub might be a jaunt though (dyke neuk?) It's also next to Shaftoe if you like a bit of sandstone bouldering.
To be honest the touristy areas are along the coast so that where most campsites are.
How long are you here for? You might be better off going closer the coast, maybe nearer Alnwick. Then you could do Simonside, Kyloe, the weird sea cliffs by dunstanburgh (bird restrictions might apply) Bowden and whatever that crag over hanging the river Coquet is. Also when the weather craps out you could go to the castles....