The sheet music in Levy, found by Masato, has seven verses, of which four are different.
1st verse: Similar, but only one watchman run-nin roun. First line of chorus as posted by Masato, above, "sings" better.
2nd verse, line 4: "....lay dar still (not laid).
3rd verse: This is verse 9 in the Marsh publication of 1854.

4: Old Dan Tucker an I got drunk,
He fell in de fire and kick up a chunk;
De charcoal got inside he shoe
Lor bless you honey how de ashes flew.

5. Down de road foremost de stump,
Massa make me work de pump;
I pump so hard I broke de sucker,
Dar was work for ole Dan Tucker.

I went to town to buy some goods
I lost myself in a piece of woods,
De night was dark I had to suffer,
It froze de heel of Daniel Tucker.

Tucker was a hardened sinner, De ole sow squeel, de pigs did squall
He 'hole hog wid de tail and all.

I hope more verses are added here. Verses changed frequently, and different minstrels made their own contributions. Several of the pioneer songs which developed from Old Dan are posted elsewhere or are in the DT.

It is an easy progression, as Sorcha says. The origin is probably in old fiddle tunes which over time have acquired a variety of verses of all kinds.

Mention of "Take Me Back To Tulsa" brings memories. At the close of WW2 I was stationed near Muskogee. There was a German prisoner of war camp there. In the prison camp kitchen, over coffee and Kaffekuchen, we helped the Germans with the songs they heard on the radio. They liked the country music on the local stations.