William and Mary's children have been identified from records available from Durham Records Online and familysearch.org.

From John (1798) onwards, the records of baptisms and burials enumerate the offspring and give both parents' full names, as well as, in some cases, William's profession and address. Records prior to the birth of John (1798) simply name the parents as "William and Mary Watson" so we cannot be certain that the earlier records relate to this couple's children. However, given that John (1798) and Mary (1805) are identified as their fifth son and second daughter respectively the following sequence can be constructed, assuming that children who died in infancy were not accounted for in the enumeration of subsequent children:

Child	Born	Died	Died in infancy?	Son no.	Daughter no.	Notes
Jane	.../07/1784				X	daughter of William & Mary Watson, born in July last.
Ann	15/02/1788	1792	Yes		X	daughter of William & Mary Watson.
John	1789-90	1790	Yes	1		son of William & Mary Watson.
Mary	.../07/1789		Yes		X	daughter of William & Mary Watson.
William	ca 1791			2		son of William & Mary Watson.
Robert	ca 1792	1792	Yes	X		son of William & Mary Watson.
Wayman	21/01/1794	1794	Yes	X		son of William & Mary Watson.
Rachael	.../05 /1795				1	daughter of William & Mary Watson.
Wayman	28/10/1795	25/04/1866	No	3		son of William & Mary Watson.
Robert	ca 1797			4		of Wearmouth, son of William & Mary Watson.
John	17/04/1798			5		5th son of William Watson (native of this parish) & Mary formerly Wayman (native of Bishopwearmouth).
Richard	12/10/1800	09/06/1802	Yes	6		6th son of William Watson ([native] of Sunderland,) & Mary Wayman (his wife, [native] of Bishopwearmouth). Burial: of Maud's Lane, son of William Watson (keelman) & Mary Wayman (his wife).
Richard	29/03/1803			7		7th son of William Watson (keelman, native of Sunderland,) & Mary Weyman (his wife, [native of] Bishopwearmouth).
Mary	15/05/1805				2	2nd daughter of William Watson (keelman, native of this parish,) & Mary Weyman (his wife, native of Bishopwearmouth).
Ann	30/11/1807	07/11/1811	Yes		3	3rd daughter of William Watson (running fitter, native of this parish,) & Mary Wayman (native of Bishopwearmouth). Burial: of Low Street, daughter of William Watson (keelman) & Mary Weyman.

Jane was born two and a half years before William Watson and Mary Wayman were married, and she doesn't figure in the enumeration of their surviving children. She may have been born out of wedlock and died in infancy, or she may have been born to another William and Mary Watson.

Mary (1789) is assumed to have died before the birth of her namesake in 1805, although I have not yet found a record of her death.

Addresses
1802: Maud's Lane,
1811: Low Street.

Occupations
1802-1805: Keelman,
1807: Running fitter,
1811: Keelman.
For another definition of a "Running fitter", see blunderingblindlybackwards.blogspot.co.uk.

The only death record I have found for a Mary Watson who could be his wife described her as the widow of Seaham Harbour Master William Watson.

Durham Records Online:
Burials, Sunderland District - Record Number: 538041.2
Location: Sunderland
Church: Holy Trinity
Denomination: Anglican
17 Apr 1845 Mary Watson, of Long Bank, age: 80, died 13 Apr 1845, widow of William Watson (harbour master for Seaham).

... in which case this is probably his death:
Durham Records Online:
Burials, Sunderland District - Record Number: 462972.2
Location: Sunderland
Church: Holy Trinity
Denomination: Anglican
2 May 1837 William Watson, of Seaham Harbour, age: 72.

My 4 x great grandfather.