In addition to Tony's advice:
Corbridge is accessible by both a regular bus (685) and train service from Carlisle and Newcastle.
Wallsend/Tynemouth easily reached by bus/metro from Newcastle.
Given the choice, I'd certainly use Corbridge as a base for your car.
Main problem with the walk is the return to Carlisle from Bowness by public transport. as there is an infrequent bus service and the taxi-bus only runs on Sundays.
http://www.hadrians-wall.org/ResourceManager/Documents/93%20Carlisle%20to%20Bowness_1_634716390232861250.pdf
If you intend stopping your last night in Bowness, then the return to Calrisle and Corbridge won't be a problem.
The HW bus is an excellent service but only runs from Good Friday to October.