There’s loads of dog walking to be had. Beamish woods that surround the museum for a start. And the general area is quite pretty, but in a very gritty northern kinda way with shot cans of super strength lager.
I’d agree with going towards Hexham. Edmundbyers, Blanchland, Slaley it starts to get quite nice (Blanchland Village is lovely, as are the surrounding walks). But anything past there is quite nice as you begin to move into Northumberland. Although you might want to stay away from the moorland with the dog. About 45 mins, or an hours drive depending how far up. Bit longer maybe if you wanted to head up towards Hadrian’s Wall around Haltwhistle area.