Well, there are a number of shantys in our copy of The Book of Navy Songs c.1955 by the Trident Society (a student organization) of the US Naval Academy. It refers back to a book called Songs of American Sailormen by Joanna C. Colcord, c. 1938, but we don't have copy of that. There is nothing in the book we have that says whether or not these songs were actually sung onboard US Navy ships during the pre-steam shanty era.

I wish you luck in proving or disproving a negative. Cyril Tawney says in the introduction to Grey Funnel Line that, "...although there is evidence to refute the oft-repeated nonsense that shantying was strictly forbidden in the Royal Navy...it is undoubtedly true that the practice was rare in the Royal service and was almost wholly confined to merchantmen." My entirely uninformed guess would be that this was even more true in American Navy, since it was extremely small until the Civil War and its geometric growth afterwards coincided with the rise of steam, the introduction of the steam winch, and the effective end of work shantys.