Learn English – Word for Secretive Marriage that is not Elope

I've heard "confirmed bachelor" for men. The only terms I've heard for women are somewhat derogatory ("old maid" or "spinster"). "Confirmed bachelorette" would be analogous but isn't common yet.

Edit based on comments: apparently "confirmed bachelor" has other connotations in some communities. I don't know how widespread that is; I'd never heard that before.

Summary

The American cut your stick, to die, comes from the British cut your stick, to depart, which dates back to at least 1813.

Americanisms

Uncle Tom's Cabin by American author Harriet Beecher Stowe was published in 1852 so let's check an almost contemporary dictionary.

Maximilian Schele de Vere's Americanisms; the English of the New World (1872) says on page 594:

Irish origin?

The OED has the British English sense of departing from 1825 but with no etymology.

The Wordsworth Dictionary of Phrase and Fable (2001) based on the original by Ebenezer Cobham Brewer (1810 – 1897) offers an Irish origin:

I must cut my stick--i.e. leave. The Irish usually cut a shillelah bfore they start on an expedition, Punch gives the following witty derivation:--

"Pilgrims on leaving the Holy Land used to cut a palm-stick, to prove tey had really been to the Holy Sepulchre. So brother Francis would say to brother Paul, ' Where is brother Benedict ?' 'Oh (says Paul), he has cut his stick ! ' — i.e. he is on his way home."

OED antedating

The OED can be antedated in Grose's Classical Dictionary of the Vulgar Tongue (1823, page 107 -- but not in the 1788 or 1796 editions) which simply says:

CUT ONE'S STICK. To be off. Cant.

Further antedatings

Further antedatings can be found in Othello-travestie: In Three Acts, with Burlesque Notes in the Manner of the Most Celebrated Commentators and Other Curious Appendices by John Poole (and William Shakespeare) (page 8):

Roderigo.

Why not cut your stick ? (b)

Page 29, just before Cassio leaves:

Cassio.

I'll cut my stick.

And in the extensive footnotes:

Given the subtitle of the book -- "with Burlesque Notes in the Manner of the Most Celebrated Commentators" -- I don't think we can trust these notes to be real, and therefore the 1597 to be fictional, but they are at least 1813 examples.