My mother actually used to sing a version of the song being discussed, its a hogde podge of all of them I think. As I can't find the one she sang spacifically.

I see the moon and the moon sees me
I'm sitting under the old oak tree
And please let the moon that shines on me
Shine on the one I love.

Over the ocean's over the seas
that's where my true love is waiting for me
oh please let the moon that shines on me
Shine on the one I love.

She had four more verses to the song, but I can't remember them--
Something about Birds and Roses, in two of them. But as it was a long time ago, I don't remember.