From the sheet music at Indiana University:


TE-NA-NA FROM NEW ORLEANS
"Words and music by The Two Robinsons" [see J. Russel Robinson.]
Indianapolis: Seidel Music Pub. Co., ©1911.

1. Dearie, let me show to you a dance.
It's not the kind that put you in a trance.
Oh, that "San Francisco Bear" is naughty, so they say,
But, honey, they don't do this dance that way!
Come up close where I can reach you,
And this lovin' glide I'll teach you.
It comes from where magnolias grow,
And balmy southern breezes blow!
It's called the "Te-na-na,"
Oh, the "Te-na-na" from sunny New Orleans!

CHORUS: Oh, that dance! Oh, that dance!
That Te-na-na, oh, the Te-na-na.
Hear that most peculiar music start?
Cuddle right up to my lovin' heart.
Trottin', trottin' all around the hall
Like a turkey gobbler at a ball!
Then toddle, toddle on the tip of your toes,
Way down south, where the Mississippi flows,
Doin' the Te-na-na, oh, the Te-na-na from New Orleans!

2. Dearie, how'd you like me for a beau?
Don't shake your little head and answer "No."
Come up closer, dearie, if you think we can agree,
And buzz sweet words of love like a honey bee.
If I thought that it could win you,
With this rag I would continue,
Till I most fell from off my seat!
Just say I've won your heart complete!
With that "Te-na-na"
Oh, the "Te-na-na" from sunny New Orleans!

[I don't quite know what to make of this. Spotify has several recordings called TEE-NA-NA by Leon Redbone, Eddie Bo, Champion Jack Dupree, Sid Morris, and Straight Whiskey with Robby Cyr, but they do not seem to be the same song. I notice there are a lot of modern New Orleans songs that contain nonsense phrases; maybe they float from song to song? I don't really have enough experience with these songs to know what's going on.]