Both texts of "The Night Before Larry Was Stretched" in Digitrad are late versions. The earliest surviving text is more pungent:

DE NITE AFORE LARRY WAS STRETCH'D

De night afore Larry was stretch'd
de Boys de all ped him a visit
bait too in dir Sacs de all fetch'd
de sweated dir duds till de ris it
for Larry was ever de Lad.
when a Boy was condemd to de squeezers
he'd swet all de duds dat he had
to help his poor friend to a sneezer
and warm his Gob fore he died.

De Boys de came crowding in fast
de drew all dir Stools round about him
nine Glims round his trapcase were plac'd
he could not be wakd well widout em.
whin one of us axd could he die
widout having truely repinted
O say's Larry dats all in my Eye
and first by de Clargy invinted
to get a fat bit for dirselves.

Im sorry dear Larry says I
to see you in dis Situation
and blister my Limbs if I lie
If I live it will be my own Station
Uchone its all over says he
de neckcloth Ill be forced to put on
By dis dime to morrow youll see
poor Larry as ded as de mutton
bekase why his courage was good.

Den Ill be cut up like a pye
and my nob from my Body be parted
your in the rong box den says I
for de never will be so hard hearted
a Chalk on de back of your neck
Is all dat Jack Ketch dare to give you
den mind not such trifles a feck
for why should de likes a dem greif you
and now boys come tip us de deck

De Cards being call'd for we pled
till Larry vount one a dem cheated
a dart ad is napper he made
de boy being easily heated
and ses be do hoky you teef
Ill splinter your skull wid my daddle
you cheat me bekase Im in Greif
but soon Ill demolish your noddle
and tip you Your Claret to drink

De gownsman step'd in wid his book
and spoke him so neat & so civil
Larry tipt him a Kilmainham look
and pitchd his big wig to de devil
den raising a little his head
he took a sup out a de bottle
and sighing most bitterly said
Oh de hemp will be soon round my throttle
and squeeze my poor windpipe to det.

But sure dis de best way to die
oh de devil a better a livin
for when on de Gallows so high
de way is de shorter to heaven
but what harrashes Larry de most
& makes his poor soul malankolly
wen he dinks on de dime dat his
Gost shall come in a Sheet to his Molly
O sure it will kill her alive

Deeze words were so meltingly spoke
our Grif it found vent in a Shower
for my part I dot my hart broke
to see him cut down like a flower
On his travels I watchd him next day
de trottler I tot to have kilt him
But Larry not one word did say
nor changed till he kem to king William
and den why his kuller grew white

When he kem to the nubbing chit
he was tuckd up so neat & so pritty
de rumbler shuot off from his feet
& he died wid his feet to the sitty
he kick'd too but dat was all pride
for soon you may say twas all over
and whin the noose was untyd at home
why we wak'd him in clover
and sent him to take a Ground sweat.

(Text from: Andrew CARPENTER, ed., Verse in English from Eighteenth-Century Ireland (Cork University Press, 1998)

+Ireland Sixty Years Ago+ (c1840?, referring to c1780?), Chapter 8 gives details of the figure who may have been the original Larry, and ventures some thoughts on the authorship of the text:

"
A man named Lambert was an outcast of a respectable family, and was known thus to have spent his last precious moments; and it was on him the celebrated song of "De nite afore Larry was stretched" is supposed to have been written. He was a cripple, paralytic on one side, but of irreclaimable habits. He was at once ferocious and cowardly, and was reported to have always counselled murdering those whom he had robbed. When on his way to execution, he shrieked, and clung with his hands to whatever was near him, and was dragged with revolting violence, by the cord about his neck, to the gallows from which he fell.

The celebrated song composed on him has acquired a lasting fame, not only as a picture of manners, but of phraseology now passed away; and its authorship is a subject of as much controversy as the letters of Junius. Report has conferred the reputation of it on Burrowes, Curran, Lysaght, and others, who have never asserted their claims. We shall mention one more claimant whose pretensions are equal to those of any other. There was at that time, a man named Maher, in Waterford, who kept a cloth shop at the market cross; he had a distorted ancle, and was known by the sobriquet of "Hurlfoot Bill." He was "a fellow of infinite humour," and his compositions on various local and temporary subjects were in the mouths of all his acquaintance …
"

http://indigo.ie/~kfinlay/60years%20ago/chapter8.htm

Carpenter notes that the last procession before the statue of King William, which stood in College Green, outside Trinity College, Dublin, took place in 1783, so that the poem must date from before this.

By 1789, the poem was already well-enough known to be parodied. See +The Sham Squire and the Informers of '98+ by William J. Fitzpatrick (1866), quoting a Dublin newspaper of 1789.):

Oh, de night afore Edgwort was tried,
De Council dey met in despair …

http://indigo.ie/~kfinlay/shamsquire/satires.htm

Robin