Most seem to. The 'Fortune My Foe' tune is particularly suitable, having been associated with all manner of tragic and lugubrious songs during its long life.

For a rather longer text with an identified source, see the late Bruce Olson's website: Scarce Songs 1

See also the broadside edition by Wright, Clark, Thackeray and Passinger at The Pepys Ballad Archive:

A sweet Sonnet, wherein the Lover exclaimeth against/ Fortune for the loss of his Ladies favour, almost past hope to get again, and in the end/ receives a comfortable answer, and attains his desire, as may here appear. To the Tune of Fortune my Foe.

Earlier copies haven't survived, but the original song was evidently late 16th century; 'The Life and Death of Dr Faustus,' registered in 1589, was set to the tune.

Other broadside songs set to 'Fortune My Foe' can be seen at Bodleian Library Broadside Ballads:

Fortune My Foe

Beware of the entries at The Fiddler's Companion for the tune: these are confused and very misleading, relying heavily on the discredited musical historian Grattan Flood, who claimed it as Irish, on the sole ground that it was on record as having been played there in 1649/50. By then, however, it was well known in many countries; in Holland, for example, it was typically known as 'Enghelsche Fortuyne' or 'Fortuyn Anglois,' and had been in print since at least 1596. (See Claude M Simpson, The British Broadside Ballad and Its Music. New Brunswick & New Jersey: Rutgers University Press, 1966, 225-231).