Jon...many folks sing these songs. Mudcatter SongBob (Bob Clayton) is part of a group which does them... and if you haven't already, check out The 2nd South Carolina String Band, who do very 'authentic' versions. (There are examples of YouTube)

Many years ago, Ernie Ford had a fine LP of Civil War songs.

Now... the naval songs are not nearly as common, so well done examples would be welcome.

(Bill... who sang "The Hippies & the Beatnicks" back at you at the Cook's in Maryland several years ago.)