Following Dale's suggestion, I found the Jack Guthrie recording on Spotify, and it's terrific. I thought I'd post my transcription of the Jack Guthrie recording:

OKLAHOMA HILLS
(Jack Guthrie and Woody Guthrie)

Many months has come and gone
Since I wandered from my home
In those Oklahoma Hills where I was born
Many a page of life has turned
Many a lesson I have learned
Well, I feel like in those hills I still belong.

    CHORUS:
    'Way down yonder in the Indian nation
    I ride my pony on the reservation
    In those Oklahoma Hills where I was born
    Now, 'way down yonder in the Indian nation
    A cowboy's life is my occupation
    In those Oklahoma Hills where I was born.

But as I sit here today
Many miles I am away
From a place I rode my pony through the draw
While the oak and blackjack trees
Kiss the playful prairie breeze
In those Oklahoma Hills where I was born.

Now as I turn life a page
To the land of the great Osage
In those Oklahoma Hills where I was born
While the black oil, it rolls and flows
And the snow-white cotton grows
In those Oklahoma Hills where I was born.

Copyright Jack and Woody Guthrie


Note the quote above in the message from John in Kansas:

It has always been my impression that Woody wrote the words and Leon wrote the Music. - Wava Guthrie Blake.