I don't know what the authorised spelling of Sarey would be. I took it off the record label. For this sort of thing, though, what appears on the label is generally what the recordists thought they heard. I've always taken it to be the (once fairly common) diminutive of Sarah.
Incidentally,I see that the BL catalogue has 'The Old Sow' as a Sussex folk song by Albert Richardson, published in 1934, which suggests that he may have been advised to get it into print after the recording proved to be a best seller.