Yes, the Navy had drinking songs. I have an old song book that is full of U.S. Navy drinking songs. The book dates from the '20's I think. I don't think it has the song retsub is looking for. I need to go home and look at it.
There is also a great album by Oscar Brand called "Every Inch A Sailor" which is drinking and bawdy songs (the two do go together) from the Navy. He did a whole series of them, one for each brnch of the service.
And then there was his series that started with "When Dalliance Was In Flower And Maidens Lost Their Heads" which at my last count was up to at least six albums.
Now how did I get into bawdy songs? I guess it's been a while....