Long reply coming up which I hope will make sense eventually.
In 1851 Dorothy has two grand-daughters with her – Elizabeth Carrick aged 11 and Mary Carrick aged 9. Looking at their birth registrations, their mother’s maiden surname is given as Armstrong. In 1841 Elizabeth aged 1 appears to be with her parents William and Mary Carrick in Gilligate.
It appears that William Carrick died in 1842 and that Mary re-married in 1845, to a Joseph Forster, with whom Mary can be found in 1851 complete with some of her Carrick children and three young children born to Mary and Joseph. These latter children’s mother’s maiden surname again given as Armstrong.
The marriage of Mary Carrick and Joseph Forster in 1845 gives her father’s name as John Graham.
Going back to the marriage of Mary to her first husband William Carrick, this was on 27th August 1831 at Hexham Abbey. One of the witnesses was a John Graham (transcription at Hexham Library).
You mention four children to John and Dorothy, but I could only find baptisms of two of them in the Hexham registers: 15th December 1816, John, son of John (labourer) and Dorothy. 28th March 1819, Ann, daughter of John (labourer) and Dorothy
However there are these two interesting baptisms in the Hexham register transcriptions, which date-wise fit with the two older children you have mentioned:
15th October 1809, Mary Armstrong bastard daughter of Dorothy Armstrong
24th May 1812, James Armstrong, bastard son of Dorothy Armstrong.
So, my suggestion is that Dorothy was actually Dorothy Armstrong, that she had two children out of wedlock before she married/lived with John Graham. Hence, when Mary Armstrong married for the second time she gave her father’s name as John Graham although in fact he was only her adoptive father.
However…..I can’t find a marriage between Dorothy and John (yet).
Hope this might help a bit.