Take a look at South Meadows Caravan & Camping Park which is open, I believe, all year round. Although a private site it is very well run by the owners.
South Meadows is 15 miles north of Alnwick and 15 miles south of Berwick (where there is a large supermarket just off the A1 --- Safeways rings a bell but it was taken over by Morrisons so might have changed names). It is about 6 miles inland from Bambrough. Although the A1 is about a mile away it was never a bother.
Dogs are welcome and there are walks from the site. The village of Belford is within walking distance and there is a golf club between the site and Belford!
Lots to see around there and the roads are quiet, which is another bonus.
I will add the website in a moment --- just going to check I have the correct one.
Ann