I have just been listening to Mateo Flecha's "La Bomba" ("The Pump"), a sort of madrigal/cantata/mini-opera from the early 16th century - he called it an "ensalada", meaning salad, and it may have been performed as a Christmas play.

It's a wild and complicated piece, describing a ship foundering in a storm. Flecha used a lot of folk material, much of it brought back by sailors from Africa or the Americas, and there are African rhythms everywhere. But they give up on pumping fairly early on and take to desperate interjectory prayers before they get rescued. You might find genuine pumping shanties somewhere in that frantic Africanized texture, but the prayers seem to me a much better bet as a source of 16th century maritime folk material.