This doesn't answer the question, but Smithsonian Magazine has an article on Shakers and Sabbathday Lake, here . There's some .wav's with fragments of shaker songs on the music page. "Simple Gifts" is one of them - the fragment is only the first verse.

I followed links from there to these pages:
Religion in Song: The Craftsmanship of Shaker Music
AMERICAN SHAKER MUSIC
150 years of 'Simple Gifts'

Also, this month's Saveur magazine has a write up on Shaker cooking.