I'd probably do Junction Road-Darlington Back Lane-Sadberge, then pick up the cycle path alongside the A66 until the end of the dual carriageway, hop over and down Haughton Road into town. Probably busy.
Alternatively, Elton-Long Newton and along the old railway line from MSG. This includes more Stockton but will be quieter on the whole, but it's a tricksy route if you don't know where to find the badly-signposted cuts away from the through roads.
Or quieter but longer - Wynard-Thorpe Thewles-Bishopton-Little Stainton-Great Burdon-Haughton Road. It should have quietened down by the time you get out of Billingham.