Joe, there is a (very) low quality recording of a video of a 78 player here: https://www.youtube.com/watch?v=1_yflcJnYnk&t=66s. Unfortunately, the second link # provided is also blocked for me.

Though it's not the best audio, her singing is clear enough that I can get it. Here's my transcription of it. There are a few question marks. I have a few uncertain marks but I'm 90% solid.

Of note, the singing is very obviously fakely and strongly accented, as Wallis was from Brooklyn, as she states in the song. Additionally, she had a penchant for 'risque' songs -- this holds true here as well. There's also a lot of good puns.

I had trouble spacing the lines logically. If you have a better way to organize them, feel free to do so.

OILMAN FROM TEXAS

I was born in Brooklyn, I was brought up in the Bronx
But mostly I have gigged around ???Manhattan??? honky-tonks
Lookin' for a feller who would make me a Cinderella
Even if he was a fat millionaire with lots of cash and not much hair
At last I found the one
My working days are done

He's an oilman from Texas, well, well, well
He's an oilman from Texas and you can tell
All he thinks about is drilling
And a girly who is willing
He's an oilman from Texas, well, well, well

You should see how he gushes over me
He wants to build me a palace way down far in Dallas
Positively no strings attached -- ain't he a liar

He's an oilman from Texas and he ain't a bit refined
But he's young and he's single and his pockets jingle jingle
And my cloth will all be silver lined -- silver lined!

He's little bit crude and he wears his pants so tight
But I know he's no dude -- haha, I know he'll do alright
Back in the saddle again --

Night and day
Day and night
Hi, ho, silver!

He's an oilman from Texas, well, well, well
He's an oilman from Texas and you can tell
All he talks of is his pipeline
It goes further, they say
Than the Atchison, Topeka and the Santa Fe
Woo, woo!

He's slick as he can be
He says he's made his mind up, he's gonna wind up
Deep in the heart of me
(He don't mean Texas!)
Deep in the heart of me
(He knows what sex is!)
And I'm inclined to say "whoop-ee!"