Back in Newcastle you could also use the Tyne and Wear Metro to go down to Tynemouth for the Priory and Castle.
Also the North Shields Fish Quay remains as an important fish landing quay, but is also a bit of a local destination for food and drink. There is also the Old Low Light Heritage Museum. That is within Cliffords Fort- which was a defensive structure built in the Anglo-Dutch Wars. The coast line around Tynemouth is quite rugged, with beaches- hence the Tynemouth Volunteer Life Brigade and their museum.
There is also the Tynemouth Aquarium
Go on a Saturday or Sunday and there is a big market on Tynemouth Station, known throughout the region.
Go to Whitley Bay on the Metro then a short bus ride for St Mary's Isle and Lighthouse, especially at dusk. This summer it is under repair so access will be limited in September and October, and the lighthouse is currently covered in scaffolding.
If you have done the Toon Tour all local buses are included in the price of that ticket. Tynemouth, North Shields and Whitley Bay/St Mary's Isle all have direct and frequent buses from Newcastle.
Unlike York, Durham, Whitby and Scarborough you are not going to be dealing with hordes of tourists in this area either.
From North Shields you can take the ferry across the Tyne to South Shields where there are more beaches, and down the seafront to the Souter Lighthouse and the Leas (a National Trust property).
Closer in to the city there is Jesmond Dene to explore.
You didn't mention Hadrian's Wall but Hexham for the AD122 bus is under 2 hours by train from York- so an early start and a lateish return gives you meaningful time on the touristy part of the wall- Housesteads, Vindolanda and the Roman Army Museum are all easy to do on a day trip (not on the same day, that would be overkill!)