and e, some minor corrections to the above, to my ear:

I WENT DOWN To NEW ORLEANS

As I went down to New Orleans
To get a glass of porter
The landlord and the landlady
Had just sat down for to eat their supper

Chorus:
Tur-a-had-lick-a-one-high-high
Tur-a-had-a-lick-a one-eyed Reilly

The girls, the girls, came flocking in
Like to bees to swarming
There was one among the crowd
And oh, Lord, God, now wasn't she charming

This pretty fair maid, whe went to bed
She left her drawers* unbuttoned
Well I knew that I might shut
The bunch of holes I found opened

I carefully pulling off my shoes
I went into her chamber
Say pretty fair maiden are you awake
And have you lodging for a stranger?

Away, be gone, replied this maid
I fear you are some rover
I hugged so sweet and kissed so neat,
Til I go her to turn over

And I fished and boxed around the hole
Till I got it open
I run my due-nick in nine inch
Never touched the bottom

And I heard a rumbling up the stairs
Who should come but her old mother
Saw me leave between her daughters thighs.
Clapped her hands and cried out murder

I quickly made a sudden spring
I did her by the shoulders
I rammed her ass again' the wall
Played the same til her own ran over **

And I heard another rumbling up the stair
But who should come but her old father
A brace of pistols in each hand
To shoot the man who fucked his wife and shagged his daughter

I quickly made a sudden spring
I seized him by the collar
Rammed his pistols up his ass
Fucked his wife and shagged his daughter

And I went out into the country
There the boys they called me a rowser
Fucked his wife and shagged his daughter
Shook my prick at his old dog Towser

* This could still be doors, as you had, rather than drawers, the undergarment. Both make sense, if you think of a spring-loaded button lock on a door (my old house growing up had this style), but I think drawers makes slightly more sense.
** I'm less sure about this verse. It's pretty mumbled.