Basil Lubbock's 1902 "Round the Horn Before the Mast," which is set in 1899, gives this complete couplet as part of "Away Rio!":
We'll sing as we heave to the maidens we leave.... You know at this parting how sadly we grieve....
Besides Mordaunt, something similar also appears in "Blue Water," by the Canadian novelist F. W. Wallace.
I don't recall seeing a similar couplet in any other song.
Whether it's related to Dana's shanty I don't know, but I do agree that if he'd heard "Rio Grande," he'd likely have used that or a similar title (like "Away, Rio")rather than a phrase in one of the couplets.
I suspect that "Heave to the Girls!" was a chorus or part of one.