OP, so much depends on your budget, if money was no object I would be looking at a house on edge of the town moor in Gosforth, 5 minutes walk to the high street (8 pubs!) loads of coffee shops and restaurants, ASDA a few hundred yards away ( admittedly about an hour by car at the moment). Walk into town about 20 mins across the moor.
Is it Newcastle or Newcastle ish? Tynemouth Monkseaton and Whitley Bay are North Tyneside, Morpeth Ponteland and Hexham are Northumberland. Be aware the coast is often a fair bit cooler than inland, might seem a great idea at the moment but it's not like this 99% of the time.
If you want similar to a London High street feel in Newcastle then realistically it's Jesmond, bits of Heaton and Gosforth. Estate agents have very loose ideas of how big Tynemouth and Gosforth are. I know you said Gosforth is too central but there is the moor between it and the city centre, most people who live there don't think of it as central.
I do quite like 5 Mile Park, wide-open ( or north Gosforth according to some) nice funky houses some with decent gardens for newish builds and right next to Gosforth Park.
We moved back to Newcastle during lockdown, dunno why we ever moved away, love the place. Gannin'yem was the best decision ever.