Although there would be fewer men available there would still be plenty (unless casualties had been high). In a normal quarter-bill (a list of where sailors would serve in particular circunstances) usually two of each gun-crew would serve as sail trimmers and would be called away from their gun by an appropriate order such as 'hands to make sail'. There was a similar arrangement for boarders (to form a boarding party or to repel boarders) and for firemen (to put out a fire).
I agree that in some circumstances shanties would have been extremely useful, such as in sailing a prize back to port (when very reduced crews were used), but I've not read any report of anything like this happening. As sailors were interchangable between the merchant fleet and the Navy, they would have been used to all the ways of merchant ships. Tradition is a very powerful thing in the Royal Navy, and it may be that it was just 'not the done thing'.