I'll check the tune in a bit to see if I recognise it, but a version of The Golden Glove does seem likely. (There are American version with With Her Dog And Gun as title). Although Tamworth is the usual location, there is a broadside in the Bodleian set in Clare: A New Song Called the Lady Went a Hunting With Her Dog and Her Gun. (At a quick glance it looks fairly standard apart from the opening).

Mick