Interesting that none of Woody's songs seem to have made it into the repertoires of those Okies, but then again Woody never exactly was an Okie, feeling quite at home in NYC. But yes, his songs do draw heavily from the Southwestern folk tradition. Here's what he wrote about his musical upbringing:

"I am pretty positive that my mother's father, Lee Tanner, was an Irishman, and that my Grandmother, Mrs. Lee Tanner, was Scottish. My mother learned all of the songs and ballads that her parents knew, and there were lots that were neither Scottish nor Irish, but Mexican, Spanish, and many made up by the Negroes in the South."

He goes on to talk about his father singing and playing blues and square dance tunes, and playing in cowboy bands. Even if we grant the fact that Woody freely embellished his story-telling to romanticize his life, this passage does suggest something of the varous hybridizing influences that have found their way into Southwestern folk music.