This doesn't really answer your origins question, but maybe it is useful and will trigger something else. I have a recording of Anchors "Away". It is on a 45 called Soldier Songs Of Our Land (Mercury Childcraft Records. MC-17). I acquired it when I was much younger. Although there is no date on neither the record nor the sleeve (quite colorful glossy coated paper) it was undoubtedly purchased in the early 1950's. I turned 3 in December of '49 and don't think I would have had a record of this kind earlier than that. To the song. The first verse is the one you posted. This is followed by a bridge and then a second verse that may be the one you alluded to. They are as follows:

Bridge:

Heave a-ho there sailor, everybody drink up while you may.

Heave a-ho there sailor, for you're gonna sail at break of day,

Drink away, drink away, for you sail at break of day.

Seond verse:

Then Navy down the field, sail to the sky

We'll never change our course, so Army you steer shy (yi-yi-yi)

Roll up the score mateys (Navy?), anchors aweigh,

Sail Navy down the field and sink the army, sink the army gray

rich r