I think Walt's tune (and text) were essentially those published in Sandburg's Songbag. I certainly began singing it about that time and may have picked it up first from Walt, and utilized his chords, too.
    Paul Clayton created a verse I like to include:

I wake up in the mornin',
Haven't got a dime.
Don't know where I'm goin',
But I'm gonna take my time,
And it looks like, I'm never gonna cease
My wanderin'.

Sandy