Quite an old thread, here is two nice lyrics pages on Bluegrass Messengers
(the author also draws lyrics from Mudcat):

Policeman (or Police , Tell Old Bill , This Morning this evening so soon ,
-Red Hot Breakdown (Earl Johnson) shares some verses, but is a much different melody.
Skillet Lickers had a tune called "Settin' in the Chimney Jamb" that is similar.

Reed Martin's version
(Oldtime Banjo in America - Kicking Mule label -out of print)

Kenny Hall and the Sweets Mill String Band
(still available as CD... and vinyl if you know where to look)

Both of these versions have altered the text or used other sources than Tommy Jarrell did.
Reed Martin's version is a little bit more strange and in modal, one of few recordings where he sings.

Here is what Tommy sang on the 'County Vinyl Down to the Cider Mill' (tune reissued
on the CD 'Stay All Night and don't go home' (rec 1967):


Police come I didn't want to go this mornin',
Police come I didn't want to go this mornin',
Police come I didn't want to go,
Shot him in the head with my forty-four this mornin'.

(Plays the tune through 3 times between the verses).

"Two little children" layin' in the bed this mornin' (2x)
Two little children layin' in the bed
One rolled over to the other and said: 'It's morning'

Bull frog jumped from bank to bank this mornin' (2x)
Bull frog jumped from bank to bank
Skinned his whole back from shank to shank this mornin'

I know something I ain't gonna tell this mornin' (2x)
I know something I ain't gonna tell...
I want to go to heaven in a coconut shell this mornin'

"Great big fellow" layin' on a log this mornin' (2x)
Great big fellow layin' on a log
Finger on the trigger and eye on the hog this mornin'

Down went the trigger and bang went the gun this mornin' (2x)
Down went the trigger and bang went the gun
Wish I had a wagon to haul him home this mornin'

(replace the quoted stanzas with the obvious words to get the original verses).

Another version is on Heritage with Kyle Creed (originally on Kyle's 'Mountain label')

A third is on Ray Alden's Field Recorders Collective, where Jarrell omits those verses (Round Peak Vol 2).

The County version is to my ears the best version, because of the energy and humor placed into it,
unfortunately much of the original verses are not good to sing in front of an audience) .

If you still want to sing all kinds of authentic and old verses many misunderstandings can occur
'if' you forget to explain the historical context of these floating verses, and what kind of times they were written for.

Rewriting:

I think really some 'floating verses' can be altered (processed) or omitted, what does matter
is your own conviction when singing "your chosen" lyrics. While it may be temporarily fun to chock any audiences or fellow band members singing the 'old minstrel and black-face show (etc) lyrics ,
you can also search up and/or write your own lyrics in the tune's tradition.
In this case, I think some kind of rhyming need to be there, and leaving out
the mention of TVs, smartphones whatever... look up similar tunes in songbooks and move verses from them to this one
as long as they fit the meter.
This is no guarantee that your version will survive,
but if the verses are good enough I would certainly think so.
If a song should be able to survive, one will have to work on it.

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Here are two verses from the Kenny Hall version:

Shoot your dice and have your fun this mornin' (2x)
Shoot your dice and have your fun
Run like hell when the police come this mornin'

Shoot your dice and roll 'em in the sand this mornin' (2x)
Shoot your dice and roll 'em in the sand
Ain't gonna work for no damn man this mornin'