The version in John and Allan Lomax, Cowboy and Other Frontier Ballads, 1938 (and later) eds., pp. 181-183, differs in some verses from the extended version offered by Rich r.
The first three verses are the same. Verse 4:

Her cruel old father did thus interfere,
Saying he would deprive her of the dearest so dear;
He would send him away where the cannon do roar
Away from his true love on the Red River shore.

Verse 5 ("She wrote me a letter ...") is the same as verse 4 in Rich r, Verse 6:

I read this letter through till it made my heart sad,
And none of my fellows could make my heart glad;
Now I'm not used to stoppin', and you may be sure,
I'm bound for my true love on the Red River shore.

Verse 7 is essentially the same as verse 5 in rich r except that the fight will be on the Red River shore (not New River; a little confusion with a genetic predecessor here). Verse 8:

I drew my six-shooter, spun around and around,
Till six men were wounded and seven were down.
No use for an army of twenty and four;
I'm bound for my true love on the Red River shore.

(Again, Red River, not New River, shore). Verse 9 is the one added by harpgirl, above. Verse 10:

Hard luck in the world for all womankind;
To those who are single the world o'er I find-
Confined with their parents until they are wives,
And stay with their husbands the rest of their lives.

These additions are occasioned by a post by Art Thieme to the thread "Info on Pretty Saro:" Pretty Saro
In an answer to his post, the version on the Max Hunter website, sung by Mrs. Lucy Quigley, Arksansas, is also recommended. The cowboy used a Winchester in her song, a more effective weapon than a six-shooter. Red River Shore