Clark 1867 (first published reference to term "chanty") mentions Shenandoah for windlass use.
Adams 1879 has it for halyards
Alden 1882 has, "One of the best known of the windlass songs was the “Shanandore”"
Dixon 1883 has "The pawls of the windlass rattled merrily to “Shanandore, I love your daughter” "
Boyd 1899 - windlass
Buryeson 1909 - windlass
Bullen 1914 - windlass and capstan

It's impossible not to be in meter, with one's body, while operating a windlass -- unless the speed of the action is so slow because you're "stuck." It naturally creates a duple/quadruple (2- or 4-beat) meter. It's a continuous action. You can't sing a chanty with prescribed time points and pull or push according to your singing. Rather, your singing must follow the timing dictated by the windlass action. A chanty at a windlass does not coordinate exertions, as it does when hauling a line. It just supports your continuous action and the strong beats of the song give an extra oomph to the exertions.

If you need a reminder of how the windlass works, see here:
https://youtu.be/IXJY_hS8SdQ

While it's theoretically not impossible to be moving one's body in meter while singing an un-metered song, it's very awkward and I would find it very unlikely. This suggests to me that there had to be performances of "Shenandoah" in meter.

Less unlikely (sorry for the awkward phrase) is to sing in a meter that is a different meter of the bodily action. The bodily action is in 4-beat meter. "Shenandoah" in 3 beats or in mixed meter (an irregular alternation between 3 and 4 beat measures) would be an example of this.

The latter, mixed meter, is the more awkward as it puts the singer-worker out-of-phase. His emphasized actions will be different each time... irregular. Some individuals (the kind of people we say "have no rhythm") would not be bothered by this, but others surely would.

The former, 3 beat metered song, would cut across the grain of the 4 beat action. The two pieces would go in and out of phase, but it would work out in the end, a sort of polymeter.

One way to avoid any going out of phase with a 3-beat song would be to subsume all of those three beats under one extortion. But this would make the tempo very slow indeed. Too slow, I think.

Most printings of 'Shendoah" have some 3-beat meter in there, whether completely in 3 or a mix of 3 and 4. It's possible we have this mix because solo singers (without chorus) sang for the transcriptions. They may have been trying to keep up the 3-beat meter but, being required also to sing the *overlapping* chorus, they had to stick in an extra beat, thus creating periodic longer measures (that writers subsequently tried to reconcile). We don't have recordings of "Shenandoah" sung by an ensemble.

To summarize so my reasoning so far: 1) "Shenandoah" was sung (not exclusively) at the windlass, and the windlass practically compels one to sing in meter 2) regular 4-beat (or you can call it 2-beat, doesn't matter) meter works easiest. As far as I know, all other windlass songs are in 4-beats (though there may be other problematic songs, like "My Dollar and a Half a Day"). 3) Yet, regular 3-beat meter or a mix of 3- and 4-beat persists in the transcriptions 4) I gravitate towards rejecting mixed meter, leaving me with (regular) 3-beat as a strong possibility as the windlass form of "Shenadoah". This is the meter used by Alden, in two examples, in what would be the earliest reliable transcription we have.

So, I move on to trying to imagine how 3-beat "Shenandoah" would be executed.

This is complicated by the timing or where syllables fall against the meter (i.e what are the strong beats/accented syllables), since these, too, are all over the place in the transcriptions. Writers did not "naturally" infer the strong syllables.