I am looking for the reference most likely in American musical theatre
of the phrase "That is my story and I'll stick to it."

Hi there,

It comes from the song "Wild About You" written by Irving Berlin, and
appearing in his musical Lousiana Purchase:

Wild About You
[VERSE:]
I'm not a McCormack, I haven't a voice
I loosen my vocals but never by choice
Still I've a song to sing
And I long to sing it for you
It isn't a ballad with plenty of heart
It hasn't the rhythm that's clever and smart
Just a tune with words that rhyme
That I wrote to say that I'm

[REFRAIN:]
Wild about you
The child about you 
Makes me wild about you
If I were a native of Borneo 
I couldn't be wilder about you

Mad about you
The bad about you 
Makes me mad about you
If I were the Hatter in Wonderland 
I couldn't be madder about you

That's my story
And I'll stick to it, dear
In case you didn't listen
In case you didn't hear

Wild about you
The child about you 
Makes me wild about you
If I were a tiger in Africa 
I couldn't be wilder about you
http://www.thepeaches.com/music/composers/berlin/purchase.html

Louisiana Purchase
------------------

Imperial Theatre, Broadway - May 28, 1940 (444 perfs) 
 
This 1940 hit musical could have been ripped from yesterday's American
headlines. Set in New Orleans, the musical lampoons Huey Long and his
strangle-hold on Louisiana Politics. When Senator Loganberry arrives
in the Big Easy to investigate the dubious doings at the Louisiana
Purchasing Company, it isn't long before he's set up for plenty of
trouble himself.
http://www.nodanw.com/shows_l/louisiana_purchase.htm


Search strategy: "my story and I'll stick to it"


Best wishes,
robertskelton-ga

Thank you so much! very informative, readable and interesting as well
as comprehensive and speedy.

There's a rather similar line in the classic 1933 film "Sons of the
Desert," in which Laurel and Hardy concoct an elaborate plot to
deceive their wives so that they can sneak off to a lodge convention.
As the deception begins to unravel, Stan Laurel says "That's my story
and I'm stuck with it."