I have the family in the 1861 census I think
821 10 page 15
18 English Street Dumfries St Marys
Peter Burns, hd, 37, woollen weaver b. Annan
Grace, wife, 32, b. Kirkmahoe
Jane, mother, 60, b. Ireland
Agnes Russell, step daughter, 10, b. Dumfries
Thomasina Burns, d, 5, b. Dumfries
James Burns, s, 3, b. Dumfries
Charlotter Burns, d, 9 months , b. Dumfries
Gadget
PS - His death certificate in 1898 does not give his parents names , so it really is searching census info, I think.