Green’s Dictionary of Slang

straw used for bedding.

c.1830		M. Moran ‘Pharo’s Daughter’ [ballad] A bull-rush tripped her, whereupon she saw / A smiling babby in a wad of straw, / She took it up and said in accents mild: / Tare an’ ages, girls, which of yez owns the child? [BS].
1835		Watford Chron. 12 Dec. 5/3: Every ‘shakedown’ in the house being put into requisition [...] each person in the house was obliged to share his ‘wad’ with a fellow-lodger.