I have "The Woody Guthrie Songbook" (1976) but the song is not in it. I learned it from "A Treasury of Folk Songs" edited by Sylvia and John Kolb (1948). It has just 3 verses and does not give any credits as to its origin. I remember a recording by Cisco Houston. It's not in John and Alan Lomax's "Best Loved American Folk Songs" though there is a song in that book called "Git Along Little Dogies" -- a completely different song, though sometimes the two songs are confused. I have two 90 min. cassette tapes of the Library of Congress recordings of Woody Guthrie done in 1940 by Alan and Bess Lomax. It may be among those songs, but there is no printed index with the tapes and I don't have the time now to listen to all of them.
I'm glad that someone pointed out that the word is "dogies", with a long letter o, and that it's not about doggies. I believe Q's explanation of the origin of the word (Spanish for motherless calf) and picked up by cowboys from Mexican vaqueros is correct.