The earliest version of the Bedlam song is, in fact, from 1615, from a manuscript book called Giles Earle, His Book in the collection of the British Library. However, the relationship of this version to the broadside is fairly distant; they do not share any complete lines. The broadside version almost seems to be written in response to the other; in the Giles Earle version, Tom seems to go mad for thinking of Maudlin, while in the broadside version, Maudlin is madly trying to get to Tom. While the choruses are different, both songs have identical metrical structures and the chorus of both begins with the phrase "yet will I sing." So it's pretty clear that there's a tradition of some kind at work. Whether the broadside author was aware of an earlier version and wrote to the same tune, or whether there were still earlier, lost versions that both are descended from, is impossible to know for sure.

Robert Graves's belief (for anyone who might be interested) was that there were earlier versions that both the Giles Earle author and the broadside author drew from; he reconstructed a "lost original" that has lines from both. But he had really no good evidence for this.

If nothing else, the Giles Earle version offers a great protective blessing:

From the Hag and Hungry Goblin That into Rags would rend ye The Spirit that stands by the naked man in the book of moons defend ye