Yarm & Egglescliffe may have been in different counties but they actually border each other - Yarm (Yorkshire) is connected to the village of Egglescliffe (Durham) by a stone bridge built in the 1400's and which still stands.
There is now a road called Yarm Road leading from Stockton-on-Tees to Yarm so I had a flick through the 1841 census and found what looks like Fawell's Buildings.
I'm not sure how familiar you are with the area but if you can get hold of a decent scale map you will see that leading off Stockton High Street is Yarm Lane, and from there you take a left onto Yarm Road, which takes you through Preston on Tees & Egglescliffe through to Yarm. From the surrounding place names on the census - Castle Buildings, Castle Gate etc- it looks likely that Fawell's Buildings would have been close to the High Street (Stockton's shopping centre is now called the Castlegate Centre).
I'm never quite sure how to reference 1841 so here is everything: Class: HO107; Piece: 317; Book: 23. Civil Parish: Stockton Upon Tees; County: Durham; Enumeration District: 1; Folio: 5; Page: 4;
Preston on Tees houses what is now Preston Park & Preston Hall Museum. Preston Hall was built in 1825 and I wondered if perhaps the lands had previously housed 'Yarm Hall'. I cant find any evidence of that but it's interesting that Preston Hall was built by
David Burton Fowler and your original link makes reference to
General Sir Fowler Burton b1822, son of David Burton.
Coincidence?
http://www.friendsofprestonpark.btck.co.uk/ParkHistory