If a marriage did take place and there seems no evidence as yet from the civil registration one did - could it have occurred after Dorothy's birth, in which case you might be looking for a differently named child?
1851 census HO107 2120 folio 104
Whitehorse Yard Gainsborough Lincolnshire
Ellen Hasthead 49 Tetford, Lincolnshire, Head Married Mariner's wife
Henry Hasthead 16 Gainsborough, Lincolnshire, Son Upholsterer (app)
Elijah Hasthead 14 Gainsborough, Lincolnshire, Son
Ellen Hasthead 5 Gainsborough, Lincolnshire, Daughter
Dorothy Hasthead 5 Gainsborough, Lincolnshire, Granddaughter
William Oxley 22 Morton, Lincolnshire, Son-in-law Married Blacksmith Journeyman
Jane P Oxley 21 Gainsborough, Lincolnshire, Daughter Married
Maryann Clements 18 Colchester, Essex, Lodger Dressmaker
In 1861 Ellen is a widow and servant in Market Rasen with her daughter Ellen. The surname is spelt Hursthead.
I actually don't really know what the surname was - as neither of these seems a surname in its own right. The birth registrations don't help for Ellen and Dorothy
Births Mar 1846
Hostkead Ellen Gainsboro 14 327
Hostired Dorothy Gainsboro 14 321
but I think the surname is probably spelt most commonly as Hosthead (but may be it is a variation of Horstead?)
Dorothy disappears after the 1851 census, though with her surname spelt in so many variations it is possible I missed finding her.
This is the only Dorothy I can see connected to Gainsborough but she is a bit of a long shot.
Regards
Valda