Worth noting, many early versions reference a Lieutenant (often spelled Leftenant) Curtis or Carter rather than the Timothy Myrick often cited as inspiration. I also searched briefly for death by snake of a Curtis or Carter whose father was a lieutenant, but turned up nothing of note.

The next earliest version I've come across was in Godey's Lady's Book, May 1840 (link). This version is quoted briefly in a story and adds little of value, other than calling it a "true, old-fashioned ballad style" song.

The Boston Morning Post of January 14th, 1841 (link) took issue with the quotation of the song in an edition of The United States Gazette a few days prior, calling it "an old poem" (quoted below). After this instance, the song/poem was quoted frequently and often in numerous papers.

* * * * *

LEFTENANT CARTER'S SON

The United States Gazette, a few days since, gave some stanzas from an old poem, in which is narrated the lamentable fate of

"Leftenant Carter's only son,
A likely youth nigh twenty-one,"

which we only notice to express our astonishment that a gentleman of Mr Chandler's learning should have been imposed upon by a spurious version of a story so famous, and to preserve the memory of the brave though unfortunate young man alluded to from injustice. The Gazetie says--

"He went on to the mounting high--
A rattle snake he did espy!
And all at once he then did feel
That p'isen critter bite his heel."

Now this is an entire misrepresentation of the facts. "Leftenant Carter's only son" did not "go on to the mounting high" --" a rattle snake he did not espy," (if he had seen the snake, of course he would have kept it at a proper distance) -- no, the true version of the story, which we copy from the earliest edition, is as follows, and we call upon the editor of the Gazette to mark its superior beauties:--

"One day this young man di-id go
Down in the meadow, for to mow;
But he had not mow'd half round the field,
When a pesky varmint bit his heeld."

The lines as we have given them have an air of consistency which renders any other guaranty of their genuineness unnecessary.

"The poet continues (says the Gazette) by describing the peculiar conduct of the victim after he had been bitten. As he went home he was heard

'Crying aloud, all as he went
Oh CRUEL, CRUEL SA-AR pent."'

Here, we are grieved to say, is another unfortunate error, which would represent the hero of an epic in the light of a whining, snivelling boy. But we have the satisfaction of knowing that young Mr Carter bore himself on the occasion more like the son of a Leftenant—

"For when he felt this varmint bite,
He raised his scythe with all his might,
And fetch-ed such a deadly blow,
That soon he laid the sarpent low."

And then, when he had thus, like a brave knight, slain his enemy, like a true one, he proceeded to lay the spoils which his prowess bad won at the feet of his ladye love." So

"He took the sarpent in his hands
And straightway went to Deacon Brand's;
Saving Molly Brand, it's here you see
The pizen creetur that bit me!"
We would give Molly's answer, if the remainder of the story was not unfortunately "torn off"