Well, other than Shenandoah the only other country-western influenced shows that I've seen are Oklahoma! and Seven Brides for Seven Brothers. I have a book that also lists the following shows under this genre (if there was an audition song listed I'll put it down):
Best Little Whorehouse in Texas
Big River "You Ought to Be Here with Me"
Paint Your Wagon "How Can I Wait" and "What's Goin' On Here"
The Robber Bridegroom
Unsinkable Molly Brown
Oklahoma! "I Cain't Say No"
I'm thinking that the show Civil War is also in this same genre and just didn't make this list. You might want to give "Missing You, My Bill" a listen. I think that's gonna be the easiest song to find from that show.
Currently~ Miss Myrna Thorn in Ruthless! the Musical
Recently~ Emma Carew (Jekyll & Hyde), Sister Mary Leo (Nunsense), Demeter (Cats), Tzeitel (Fiddler on the Roof)