Looking at the Traditional Ballad Index-
"The Dying Cowboy" in Belden, pp. 397ff., are versions of "Bury Me Not on the Lone Prairie."

"The Dying Cowboy" in Randolph, "Ozark Folksongs," is the same as "The Dying Ranger," as posted by Joe, from Doerflinger. Randolph gives two versions, coll. 1928 and 1938. He compares it with "The Dying Soldier," which is very similar, coll. in 1928.
Randolph says another was "called "The Dying Woodsman," about a Texan who was killed in the Spanish-American War, was copyrighted by J. D. Patton in 1904 and published in "The Sunny South Quartet Book (Dalton, GA, 1912, No, 25). See also the "Dying Ranger" song published by Lomax ("Cowboy Songs," 1910, p. 214), Cox ("Folksongs of the South," 1925, p. 263) and Finger ("Frontier Ballads," 1927, p. 170)."

The song in Lomax (1910 and later printings) included music. It seems to be the song copyrighted by Patton in 1904, or close to it, but I have not seen the Patton song.
Lomax did not comment on the source of the song.

THE DYING RANGER

The sun was sinking in the west
And fell with lingering ray
Through the branches of a forest
Where a wounded ranger lay;
Beneath the shade of a palmetto
And the sunset silvery sky,
Far away from his home in Texas
They laid him down to die.
2
A group had gathered round him
His comrades in the fight,
A tear rolled down each manly cheek
As he bid a last good-night.
One tried and true companion
Was kneeling by his side,
To stop his life-blood flowing,
But alas, in vain he tried.
3
When to stop the life-blood flowing
He found 'twas all in vain,
The tears rolled down each man's cheek
Like light showers of rain.
Up spoke the noble ranger,
"Boys, weep no more for me,
I am crossing the deep waters
To a country that is free.
4
"Draw closer to me, comrades,
And listen to what I say,
I am going to tell a story
While my spirit hastens away.
Way back in Northwest Texas,
That good old Lone Star state,
There's one that for my coming
With a weary heart will wait.
5
"A fair young girl, my sister,
My only joy, my pride,
She was my friend from boyhood,
I had no one left beside.
I have loved her as a brother,
And with a father's care
I have strove from grief and sorrow
Her gentle heart to spare.
6
"My mother, she lies sleeping
Beneath the church-yard sod,
And many a day has passed away
Since her spirit fled to God.
My father, he lies sleeping
Beneath the deep blue sea,
I have no other kindred,
There are none but Nell and me.
7
"But our country was invaded
And they called for volunteers;
She threw her arms around me,
Then burst into tears.
Saying, 'Go, my darling brother,
Drive those traitors from our shore,
My heart may need your presence,
But our country needs you more,'
8
"It is true I love my country,
For her I gave my all.
If it hadn't been for my sister,
I would be content to fall.
I am dying, comrades, dying,
She will never see me more,
But in vain she'll wait my coming
y our little cabin door.
9
"Comrades, gather closer
And listen to my dying prayer.
Who will be to her as a brother,
And shield her with a brother's care?'.
Up spake the noble rangers,
They answered one and all,
'We will be to her as brothers
Till the last one does fall."
10
One glad smile of pleasure
O'er the ranger's face was spread;
One dark, convulsive shadow,
And the ranger boy was dead.
Far from his darling sister
We laid him down to rest
With his saddle for a pillow
And his gun across his breast.

Pp. 214-218, "Cowboy Songs, 1929 printing. No additions in the 1938 edition, but the score is better.