There's no question that the tune of "Turkey in the Straw" developed from that of "Old Zip Coon," which - at least in its original publication - is less graceful but clearly the same tune.

In fact, the version of Zip printed by Elias Howe in his 1843 "Complete Preceptor for the Accordeon [sic]" is identical in all but title to "Turkey in the Straw" as usually played today.

Zip is usually attributed to G. W. Dixon (1834), but the Mississippi Journal (Natchez) (Nov. 23, 1832) mentions its performance by blackface minstrel Bob Farrell. According to the Nashville Daily Advertiser (Nov. 1, 1833), Farrell later performed it in Nashville with lines about David Crockett - who was in the audience by invitation.

The tune's later title of "Natchez Under the Hill" appeared by 1839 (in the variant "...on the Hill") in George P. Knauff's collection called simply "Virginia Reels." (Thanx for this bit of info to Andrew Kuntz's "Traditional Tune Archive":   

https://tunearch.org/wiki/Annotation:Natchez_Under_the_Hill_(1)

Natchez Under the Hill was a steamboat landing that was also the red-light district of Natchez, which loomed over it on a bluff. The district was mostly destroyed by fire in 1836.

Just when and where the tune came to be called "Turkey in the Straw" isn't clear, but according to James Fuld, on July 12, 1861, blackface minstrel Dan Bryant received a copyright for his song "Turkey in de Straw." Both music and words were new, but the familiar "Zip Coon" air (without words) was included at the end under the heading "Old melody."

I haven't seen any of the familiar versions of the words in print before the 20th century.

An 1882 printing of the chorus only gives it as:

Den a turkey in a straw, den a turkey in a straw,
Den a turkey in a straw, den a turkey in a straw,
Roll a web a straw round to hide the turkey's paw,
And we'll shake 'em a tune called turkey in a straw.

Finally, it looks to me as though an early source of the tune is likely to have been the slow Scots air "The Bonnie Black Eagle," in print in 1762:

https://tunearch.org/wiki/Bonnie_Black_Eagle_(The)

In other words, the "Old Zip Coon" melody may have been played for more than half a century before Bob Farrell under a different - and possibly lost - title.