That-a-way is of British origin and dates from the mid-17th century.
Here's an example from a footnote to Richard Brome's "A Jovial Crew; or, The Merry Beggars" in The ancient British Drama (1810):
28 Skise out this away, and skise out that away. — I should suppose we ought to read
" Skir out this a way, and skir out that a way."
To skir is to scour, to pass hastily.
The same play with this same line was also published in 1744 in A select collection of old plays:
.. out again at afternoon, and so 'till supper-time; skise out this away, and skise out that away ; (he's no snail I assure you;) and Tantivy all the country over, where hunting, hawking, or any sport is to be made, or good sellowship to be had ; and ...
In fact, the comedy was first staged in 1641 or 1642 and first published in 1652. Samuel Pepys saw it and wrote in his diary:
... and so I went away with Mr. [Henry] Moore, and he and I to the [King's House] Theatre [Royal], and saw “The Jovial Crew,” the first time I saw it, and indeed it is as merry and the most innocent play that ever I saw, and well performed.
This side-by-side webpage of the 1652 quarto and the modern text includes this on the old side:
... Skiſe out this a-
way, and ſkiſe out that away. (He’s no Snayle I as-
And for the modern:
Out again at afternoon, and so till supper–time. Skice out this– a–way, and skice out that–a–way.* — He’s no snail, I assure you.
The asterisk notes:
] This edition; Skise out this away, and skise out that away. Q1. OED wrongly identifies 'this-a-way' and 'that-a-way' as nineteenth- and twentieth-century Americanisms indicating direction. See this-a-way, adv., 2, and that-a-way, adv., 2, although the second citation for skice, v.1, is this line in Brome, meaning move quickly; skip or frisk about.