I don't believe any seamen sang it at sea in the 19th century. It's too beautiful, lyrical, emotional - in word, sappy - for seamen. Any seaman singing that on board would have been thought to be playing for the other team.

And Taconicus is right, it's pronounced Shenn-ann-doh in the song. If you don't want to look like a dweeb, sing it that way.   

A geographical name, such as Shenandoah Valley, is pronounced Shenn-ann-do-uh.

It is futile to argue about pronunciation involving R. Even today, with education and movies, English speakers from different areas handle R differently. Some drop it, some trill it, some voice it. Some put it where it doesn't belong (arear for area) and some leave it off where it does belong (carpentuh for carpenter).

Claiming that one R or another is authentic is just plain silly.