I found it. Willie's recording is on "Red Headed Stranger," which many think is Willie's best album. Willie's rendition has a tune that sound to me just like "Red River Valley."

Can I sleep in your arms tonight, lady?
It's so cold lying here all alone
And I have no hope (?) to hold on you
And I assure you I'll do you no harm.

I don't think it's the best song on the album, but it's OK.
-Joe Offer-