Here's the full chorus from the George M. Cohan song. (It turns out that Page 4 exists; you just can't get to it from Page 3, but you can get to it from the search results page.)

CHORUS: Oh! Mister Moon, my baby, Oh! Mister Moon, my love,
Oh! Mister Moon, if I had a balloon,
Well, I'd cert'nly sail to realms above, love.
Oh! Mister Moon, my sweet thing, look down upon this coon.
I am so lonely if you will only say you will be my Moon.

[Then follows an 8-bar instrumental section called "Dance."]

Getting back to the song originally requested:

I've found lots of "folk" versions of this song on the Internet. Sometimes it's shiny moon, shining moon, shinin' moon, silvery moon, or silv'ry moon. Sometimes it's even Mister Sun, not Moon. Often it's classified as a kid's song, scout song, campfire song, etc.

I have yet to find any site that gives an author or date, let alone convincingly old or complete (in my opinion) lyrics.

Related threads:
Bright and Shiny Moon
Lyr Req: Mr. Moon