Sometimes it takes a while, and sometimes you wind up doing it yourself. Found it today in Folio of Songs by "Slim" Bryant and his Wildcats, Vol. IV, 1940.
DREAMY GEORGIANA MOON
Words and Music by Clayton McMichen & Hoyt "Slim" Bryant, copyright 1935.
Dreamy Georgiana moon,
No one was there but you,
When I looked into your eyes,
With a sigh, whispered I would be true,
Each night brings sorrow to my heart,
Tell her that I'll be returning soon,
My love's still true, though we're many miles apart,
Tell her dreamy Georgiana moon.
I have put up the sheet music from Bryant's book, first page here and second page here. Thanks to Stewie for giving me a shove in the right direction.