The only place that I can think of that ticks all those boxes is Alnmouth – a nice mix of everything.
I'd avoid Amble, it a working town really.
Warkworth is a nice village with a modest range of shops and eateries.
Inland Alnwick is the obvious choice.
Seahouses is nice enough, and probably has everything you need, but not what you would call picturesque.
The whole area isn't very commercial (which may or may not be a bad thing for you), and if you don't like visiting castles other sorts of attractions are thin on the ground. If the weather is shite you'll struggle to entertain the kids for a fortnight.
Great beaches and very relaxing though!