Oh yes, Beamish Museum is great!
OP, you’re getting a weirdly firm steer away from Newcastle here, but I think the people insisting you go to York instead are being quite narrow in what they regard ans interesting/worth seeing. Both places are good for a couple days’ sightseeing, though they are quite different cities. York is historic and pretty, but will be rammed in tourist season, especially around The Shambles and the Minster.
Newcastle is buzzy and post-industrial - also historic, but less “kings & bishops” and more British industry and regeneration, with arty neighbourhoods like Ousbourn and a city centre with lots of shops & restaurants. The Quayside area (river side) has a great walkway and there are usually markets and beer gardens along it during the summer. As someone else has mentioned, the coast at the mouth of the Tyne river is lovely and easily reached by metro - go to Tynemouth or Whitley Bay to enjoy the seaside and the markets. And if you want a dose of historically pretty, charming cathedral city & castle as well, you can get a train to Durham and be there in 10-15mins. (The cathedral is where the Venerable Bede is buried, if that sort of history appeals to you!)
Heading north from Newcastle by train, Morpeth is a lovely town with lots of independent shops. And further up the coast, Berwick-on-Tweed is a little run down these days but a fascinating day trip. It was a wealthy military town during the Regency and the architecture is impressive. It also features a very old bridge that James I was reputed to have crossed when travelling south for his coronation. Not somewhere to stay, but worth seeing if you’re at all into history.
The Northumbrian coast is gorgeous but my main concern would be transport. We did a week in the NE a few years ago as a family but we stayed in Seahouses and rented a car to explore the beaches, Bamburgh Castle, Lindesfarne (Holy Island), Craster, Dunstanburgh Castle, etc. You’ve had a couple of people chip in to talk about buses, but I think if you’re based in Newcastle it could be quite tiring shelping up & down the coast on buses. Most of the pretty coastal towns and villages aren’t on the train line, sadly. If you did have a car even more beautiful spots would open up - Alnwick & its castle is also well worth the visit.
So: I would say that I’d you’re very strictly sticking to public transport, you could split your time between Edinburgh, Newcastle, Morpeth, Berwick-on-Tweed, Durham, and York and get a good sense of the region and its role across several points in history. If you could drive around the coast though, you’ve have more than enough to see just sticking to Newcastle & surrounds.