There are quite a number, so the only real answer is that the "correct" tune is the one to which the particular set of words concerned were sung at the time of recording or transcribing from tradition (obviously some variation is to be expected). Presumably your tune was originally found that way; you might find it interesting to trace it back to its source, though if you just learned it down the pub from somebody who didn't say where they got it, that might be difficult.

On the other matter, the song seems to belong to the early-to-mid 18th century. Early printings include Herd, Ancient and Modern Scottish Songs, II, 1776, p 2; and Johnson, Scots Musical Museum II, 1788, p 118 (No.115). A tune called The Lowlands of Holland was published by James Oswald c.1742 (A Collection of Curious Scots Tunes, and perhaps also c.1745, Caledonian Pocket Companion); it seems first to have appeared in the early 17th century as My love shoe winns not her way (Skene MS).

The song may have been based in part on a broadside ballad, The Seaman's Sorrowful Bride, printed in London for J Deacon, Guilt-spur-street, c. 1683 (see Ebsworth, Roxburghe Ballads VI (part 17, 1887), pp 444-5). The tune prescribed for that was Ah! Jenny gin (so named from its use with a song in Aphra Behn's play of 1682, The City Heiress).

Child includes Herd's text of Lowlands in his introduction to Bonny Bee Hom (Child 92) because it contains some of the same material; but doesn't suggest that the songs are otherwise related. Perhaps Lowlands also borrowed verses from Bee Hom, but that's conjectural. Bronson prints no tune for Bee Hom, but several for Lowlands (Bronson II, 418, no. 92, appendix), dividing them into two main groups, both apparently descended from early 17th century tunes. His summary of the song's history is probably as good as you'll get.