South Coast is also not traditional.
You can pretty much run through the 305 English and Scottish Popular Ballads and skim off the saddest 100, and you have 100 sad traditional songs. And there are 7000 versions of those 305 balads in Bronson, many of which could be called distinct "songs," so if you can read music (I can't) it should not be hard to come up with a couple thousand sad traditional songs in the time it takes to read it through.

My pick for saddest would be "The Clerk's Twa Sons of Owsenford," which is a lovely song, although it is not often sung.