I have a set of tapes of single tips from 1950 by Rickey Holden and Cal Golden. In one of the Golden tapes Cal uses the term Taw referring to partner. Clearly our use of the term has been around for a long time.
 
Jim Mayo
 
In a message dated 7/20/2017 4:47:00 P.M. Eastern Daylight Time, trad-dance-callers@yahoogroups.com writes:
 


The etymology of the term "taw" to mean a square dance partner is
a topic that I've seen come up before--I think both on the
rec.folk-dancing Usenet newsgroup and on some MWSD list I used to
subscribe to--but never with any definitive resolution.

One explanation sometimes offered is the one alluded to by Phil
Jamison and Richard Hart, namely, that it derives from use of
"taw" to mean a favorite marble. However, I haven't seen any
citations of early sources to support this idea. I've only seen
it offered as a conjecture after the actual origin of the usage
was already obscure.

The word "taw" also occurs in the phrase or "bring [someone] to
taw" (or "come to taw"). The precise nuance of meaning implied
by this phrase isn't clear to me, but it seems related to the
ideas of getting someone under control and making them toe the
line, and perhaps by extension the idea of bringing ("dragging"?)
someone to the altar.

You can find a long discussion of the phrase "come to taw" on
the Mudcat Cafe site here:

http://mudcat.org/thread.cfm?threadid=50284

One of participants quotes the Oxford English Dictionary (1989
edition) as citing this poem from a 19th-century source:

He smiles at all the girls he meets,
And you smile at him on the crowded streets,
Why don't you make him 'come to taw',
I know he wants a mother-in-law.

Did this sort of usage "come/bring to taw" lead to the use of
"taw" to mean a square dance partner? Perhaps. But the idea
would seem more compelling if someone could provide evidence of
any regional usage of "taw" to mean a spouse/sweetheart outside
the context of square dancing.

(By the way, I have no idea whether the expression "bring/come
to taw" is etymologically connected with the phrase "toe the
line" or with the "taw line" in marbles games.)

Another suggestion is that "seesaw your taw" is a mangling via
oral tradition, of the French phrase "chasse autour' (sidestep
around). I think I first read that idea in an article by Stewart
Kramer, which you can find here:

https://www.ceder.net/ArticleCoOp/articles/sashay.php

Personally, I think that this explanation has more of a ring of
plausibility than the others. What keeps me from being fully
sold on it is that I haven't yet seen an example of the phrase
"chasse autour" (or the plural/polite form "chassez autour",
with implied subject "vous" instead of "tu") appearing in an
old French dance manual. I'd be interested if anybody can cite
one.

--Jim