It seems to me that Sing Out (and by extension Rise Up Singing) have mistakenly attributed the words the Carter Family sing to Texas Gladden.

As of the time of this posting, here's the Carters singing the words from SO and RUS:
https://youtu.be/1l_zYH_YYnE

And here's two versions of Texas Gladden singing the song, under the title "I Never Will Marry" and "I'm Never To Marry."
https://youtu.be/AANPF9b7wAY

The words are almost exactly the same in both versions and quite different from the Carters/SO/RUS. Here's how I catch them:

One morning I rambled down by the seashore
And the wind it did whistle and the waters did roar
I heard some fair maiden give a pitiful cry
And it sounded so lonely, it swept off on high

Said William to Molly, if you will agree
To give your consent, love, to marrying me
My mind is to marry and never to part
For the first time I saw you, you wounded my heart

Go away from me William, and leave me alone
For I’m a poor girl, and a long way from home
I’m never to marry, nor be no man’s wife
I intend to live single, all the days of my life

The shells in the ocean shall be my deathbed
While the fish in deep water swims over my head
I’m never to marry nor be no man’s wife
I intend to live single all the days of my life

He cast her fair body in the water so deep
And she closed her pretty blue eyes forever to sleep
You promised to marry, you blighted my life
Now the girl that I hated is being(?) my wife


I suppose it's possible Texas Gladden sang a third version that followed the Carter Family lyrics, but it strikes me as unlikely.

The Kossoy Sisters seem to follow Texas until the end, when they swing back to Carter Family lyrics, probably because the order and/or pronouns at the end of Texas' version seem to get a little garbled.

As someone that's played "Never Will Marry" for a few years, I'm a little gobsmacked by this version. I think it's so much cooler! Instead of a lovesick girl melodramatically killing herself over lost love, it's a woman refusing marriage and saying she plans to live single the rest of her life (a line that never made sense when she was about to commit suicide), which provokes her failed suitor to kill her. (Shades of Pretty Polly)