There are several libraries that have the sheet music. They do not make the images viewable online due to copyright restrictions, except for the cover, which you can see at the website of Mississippi State University. It is cataloged thus:

THAT LITTLE BOY OF MINE
Words and music by Benny Meroff, Wayne King & Walter Hirsch, ©1929.

First Line of Song: Who cares for fame and fortune,
First Line of Chorus: A tiny turned up nose, Two cheeks just like a rose,

There were early recordings by:

1931 – Adrian Schubert and His Orchestra
1931 – Ben Bernie and His Orchestra
1931 – Fred Jeske
1931 – Jerry Fenwyck and His Orchestra
1931 – Jim and Marion Jordan
1931 – The Deckers
1931 – The Melodears
1931 – Will Osborne and His Orchestra
1932 – Bert Lown and His Biltmore Orchestra
1933 – The Vagabonds
1935 – Dean Boys
1936 – Jack Savage and His Cowboys
1938 – Green Brothers Marimba Orchestra
1947 – Rex Allen & His Arizona Wranglers
195? – Gene Autry
1950 – Larry Vincent
1951 – Patsy Montana
1951 – Red Foley
1951? – Eddy Arnold
1954 - Ray Haney with the Rio Grande Playboys
1955 – Wesley Tuttle
1956 – Ferlin Husky
1958 – Burl Ives
1960 – The Browns
1965 – Bob Jennings
1968 – Archie Campbell
1970 – Bonnie Owens

THAT LITTLE GIRL OF MINE
1938 – Julian Johnson & Leon Hyatt

I found a “fan” website devoted to this song: http://thatlittleboyofmine.com/. It has the following lyrics:

VERSE 1: Who cares for fame and fortune?
Who cares for wealth or gold?
Because I find a fortune,
Within my arms I hold.

CHORUS 1: A tiny turned up nose,
Two cheeks just like a rose,
So sweet from head to toes,
That little boy of mine.
Two arms that hold me tight,
Two eyes that shine so bright,
Two lips that kiss “Good-Night,”
That little boy of mine.
No one will ever know
Just what his coming has meant,
Because I love him so,
He's something heaven has sent.
He’s all the world to me.
He climbs up on my knee.
To me he'll always be
That little boy of mine.

VERSE 2: I dreamed about a castle.
’Twas high up in the skies;
But in a little cradle,
My dreams I realized.

CHORUS 2: A tiny turned up nose,
Two cheeks just like a rose,
So sweet from head to toes,
That little boy of mine.
Two arms that hold me tight,
Two eyes that shine so bright,
Two lips that kiss “Good-Night,”
That little boy of mine.
And when he lays his head
Upon his pillow so white,
I pray the Lord above
To guide him safe thru the night.
In dreams I see his face
And feel his sweet embrace.
There’s no one can replace
That little boy of mine.

And the owner has posted images of the sheet music in two parts: here and here, and has also posted abount 20 recordings.

The earliest recording I can find online is this one at the Internet Archive by Bert Lown and His Biltmore Orchestra.