Hey, Giac. Started this thread when I was half-asleep, and just now remembered to check back on it. No wonder I can't remember who this guy was... ;)
Well, (the late) Lonnie Pitchford sounds a LOT like the music I heard all those years ago. I checked out some samples of his CD, "All Around Man," and I think the song I heard was a version of "My Babe." (My memory is coming back a bit...) BUT, I'm sure the one I heard was a field recording. I (now) remember the musician on the radio talking after the song, responding to questions about the song, and saying the name of the song was "My Babe." And I do remember him sounding very old. (Though as we've seen, my memory sucks.)
Whether it was Mr. Pitchford or someone else, thanks so much for your help, and for introducing me to Mr. Pitchford's music. (But I'm going to keep looking for that field recording, too...) :)