There are several broadside copies of Digging For Go[u]ld at  Bodleian Library Broadside Ballads.  No tune is indicated, but it's worth mentioning that broadside copies of The Humours of Donnybrook (the one that starts "Here I am boys a sample of precious bad luck, not "To donnybrook steer all ye sons of Parnassus", which latter is also known as Donnybrook Fair) name Digging for Gold as tune; the jig The Humours of Donnybrook is also known as The One (or Old) Horned Sheep, and may be found under those titles at  J C's Tunefinder.