As you noted, there is an Andrew COX baptism to Andrew and Elizabeth COX
on the IGI, at Stoke Damerel, 23 December 1798. He was born 15 July 1798.
I presume your brick wall is not being able to find a marriage for
Andrew and Elizabeth?
There is a clue to Elizabeth's maiden surname, in that a Jane Brimble
Cox was baptised to Andrew and Elizabeth, 17 July 1796 at Stoke Damerel.
They also baptised at Stoke Damerel:
James Cannon Cox, 26 October 1794, born 30 July 1794 (Cannon was
possibly Andrew snr's mother's surname). There are quite a few CANNON
baptisms at Stoke Damerel in the 1700s.
So you are looking for a marriage, probably pre 1794 unless they
baptised chn elsewhere before that.
Might help with your search... hopefully!