I'm deeply suspicious of the proposed connection between Our Captain Cried (Roud 602) and The Bold Privateer (Roud 1000). Well, actually I don't believe they're related at all. The Traditional Ballad Index has taken Gale Huntington's word on it and (it isn't the only case) I think he was wrong. We don't know what his reasoning was, unfortunately, so it's hard to disprove the suggestion other than by pointing out that the two songs have virtually nothing in common structurally, and that nobody else seems ever to have suggested any relationship. There were many hundreds of songs on similar topics, so the basic plot is no guide.

More convincing is Peter Kennedy's suggestion that Our Captain Cried is really A Blacksmith Courted Me seen from the male protagonist's perspective. There is at least some supporting evidence (there doesn't appear to be any for Huntington's assertion) in that the two songs share the same metre, and occasionally verses and even tunes. Kennedy prints a set of Blacksmith from Phoebe Smith (Folksongs of Britain and Ireland, 346 and 370) which includes two verses from Our Captain Cried. It's equally arguable, however (and that's my feeling so far), that the structural correspondences between the songs are what led to the tune and text transfers between them, not any common ancestry.

The metre, relatively unusual nowadays, was not uncommon in broadside songs of the 17th century, for one thing; as were elements appearing in Our Captain Cried, come to that. Perhaps the most likely ancestor of the 19th century broadside (printers frequently re-issued or re-wrote older songs) is a pair of songs, The Seamen and Souldiers Last Farewel to their Dearest Jewels (Pepys Collection) and its sequel, The maiden's lamentation. Or, An answer to the seamen and souldiers last farewell to their dearest jewels.

Final two verses from the first:

Hark how the Drums do beat
with Trumpets sounding,
Souldiers in furious heat,
Foes would be wounding:
From thy sweet company,
although it grieves me,
I must devided be
and forc'd to leave thee.

My Captain ca[l]ls* away,
in hast they hurry,
To march without delay,
I may not tarry:
Patiently thou must bear,
love leave thy weeping,
Farewel my dearest dear,
till our next meeting.

* not entirely clear in the black-letter. Could possibly be "sails".

Second verse of the sequel:

Why wilt thou cross the Seas,
to fight with strangers,
When thou mayst live at ease,
free from all dangers:
I'le fold thee in mine arms,
nothing shall grieve thee;
I'le keep thee from all harms,
dear do not leave me.

There are other correspondences in the song. Both the above were from the same printers (see link above to Bodleian Library Broadside Ballads). Another verse seems to derive from The Kind Virgin's Complaint against a Young Man's Unkindness:

Ladies, take my Advice,
You have rare Features,
Always be coy and nice
To such false creatures;
No Man will constant prove,
No not my Brother,
Then if you need must love,
Love one another.

All three songs prescribe the same tune, Cupid's Courtesy. I think that it's ancestral to one of the tunes found in tradition with Our Captain, but the metre may be a factor in that feeling, of course. It has led to confusion elsewhere, as I've suggested.

Meanwhile, I really do believe that The Bold Privateer is irrelevant to the discussion. Steeleye Span's dog's-breakfast collation, which also includes bits they wrote themselves, is best ignored in that it tells us nothing about the history of the song under discussion and is likely to mislead the unwary. It's also a pity that none of the examples of Our Captain given so far quote sources (any text from Sedley is likely to be a collation from several different places; not even always from related songs). I may have identified some in earlier discussions, but will look at them further.