I agree with those who say, acknowledge your source if the situation makes it possible for you to do it reasonably smoothly. Saying something like, "These are words that I wrote; the tune is my adaptation of an old folk tune called The Cruel Brother who Challenged Robin Hood and the Elfin Knight to Hunt the Hare in the Forests of Van Diemen's Land" doesn't take too long, and at least some audiences are likely to appreciate a short spell of background information.

Giving the tune a new name isn't necessarily an act of deliberate obscuratanism. When Ralph Vaughan Willaims named his variant of Lazarus/Star of the County Down/Maria Martin/Come All You Faithful Christians/We are Poor Frozen-out Gardeners/ Gilderoy he was just adding yet one more name to an already long list. Kingsfold, in Sussex, was where he collected a variant of the tune, to the words "Maria Martin", from a Mr. Brooker, at theWheatsheaf Tavern, on December 23rd, 1904. (Is that tavern still there ?) See the Journal of the Folk Song Society #7, p. 118-119 (1905).

T.