If One Won't Another One Will is known as "Lonesome (Stormy) Scenes of Winter, The" [Laws H12]. The song was collected by Belden in 1904. It's referenced to October, 1887 by Meade and appeares in Wehman's Collection of Songs.

If anyone has either the Belden or Wehman lyrics please post them. Here is a list of collections:

Belden, pp. 195-196, "The Lonesome Scenes of Winter" (1 text)
MHenry-Appalachians, pp. 136-137, "The Lonesome Scenes of Winter (All in the Scenes of Winter" (1 text)
Wyman-Brockway II, p. 94, "The Gonesome [sic] Scenes of Winter" (1 text, 1 tune)
Dean, pp. 108-109, "Lonesome Hours of Winter" (1 text)
Fowke/Johnston, pp. 156-157, "The Stormy Scenes of Winter" (1 text, 1 tune)
Fowke/MacMillan 57, "The Lonesome (Stormy) Scenes of Winter" (1 text, 1 tune)
McNeil-SFB1, pp. 127-129, "Lonesome Scenes of Winter" (1 text, 1 tune)
Shellans, pp. 38-39, "The Scornful Lover" (1 text, 1 tune)
Creighton/Senior, pp. 209-212,"Stormy Winds of Winter" (4 texts, 1 tune)
Creighton-SNewBrunswick 50, "The Stormy Winds of Winter" (2 texts, 2 tunes)
Peacock, pp. 445-446, "Flora" (1 text, 1 tune)

Some recordings:

Lewis McDaniel & Walter Smith: "I Went to See My Sweetheart" (Victor 23505, 1930; on ConstSor1)
Southern Melody Boys, "Lonesome Scenes of Winter" (Montgomery Ward 7227, 1937)

IF ONE WON'T ANOTHER ONE WILL- Carter Family 1932

I went one Sunday evening
My true love for to see
I asked her to marry
And she would not answer me [INSTRUMENTAL BREAK]

The night is almost spent
It is nearer the break of day
I'm waiting for an answer
Oh, what will you say [INSTRUMENTAL BREAK]

Kind sir, if I must tell you
I'd choose the single life
I never thought it suited
For me to be your wife [INSTRUMENTAL BREAK]

In the course of six weeks later
This lady's mind did change
She wrote me a letter
Kind sir, I feel ashamed [INSTRUMENTAL BREAK]

I feel as though I slighted you
I cannot hear you mourn
So here is my heart, come take it
And claim it as your own [INSTRUMENTAL BREAK]

I wrote her back an answer
And sent it back in speed
I own that once I did love you
I loved you dear indeed [INSTRUMENTAL BREAK]

But then my mind has changed me
I seek another way
Upon some pretty fair maiden
My heart will have its sway [INSTRUMENTAL BREAK]

Upon some pretty fair maiden
My heart shall have its fill
This world is wide and lonely
If one won't another one will