Sagebrush Sadie was the title of a 1928 Disney cartoon featuring Oswald the Lucky Rabbit (he rescues her). Pete was the villain. There were 26 Oswald cartoons before Disney lost the rights and created Mickey Mouse.
I would be very surprised if a song didn't originate from this cartoon- and I vaguely remember one, but at my age and remove from that time, my mind may be playing tricks on me. The cartoon was silent, so the song couldn't have come from the film.