Hi Matt,
Ref for 1851 is HO107 2225/297 Page 6, 58 Geo Leigh Street, Ancoats:
Harriet GADD, Head, W, 53, Fustian Cutter b Lancs Salford
Robt GADD, Son, U, 21, Picker Maker b Lancs Manchester
Louisa GADD, Dau, U, 14, Womans Cap Maker b Lancs Manchester
Robt GADD, Uncle, Wid, 77, Pauper Bricklayer b Ches Nantwich
1841, HO107 576/8/33 Page 8, Geo Leigh Street, Ancoats:
Harriet GADD, 40, Fustian Cutter, Y
Maria GADD, 20, Y
Robt GADD, 10, Y
Louisa GADD, 4, Y
Baptism of Louisa at Manchester Cathedral 25 Dec 1836 shows parents as William (Fustian Cutter) and Harriet of Manchester.
I think that the key is to find the marriage of William and Harriet. As she was born in Salford according to the 1851 (is this consistent in 1861?) that may be where the marriage took place. It might be worth a separate post on the Lancashire board to see what records are available.
There is also this:
Baptism William GADD 12 Aug 1793 Gravel Lane Wesleyan Methodist, Salford born 8th June 1793 parents John GADD & Hannah - there are a number of other GADD children to the same parents.
The death certificate might help as it should give you William's age at death which would help to pinpoint him, however I can't see a death in the right area in 1837 on freebmd - where have you found it?
Jean