Green’s Dictionary of Slang

a walk that is casual and yet confident; occas. attrib.

1900		Ade More Fables in Sl. 184: Lutie never got out of her Dream until she made a bold Sashay with a Concert Company.
1920		Ade Hand-made Fables 178: He is thinking that probably he had better do a quick Sashay into Wall Street, fill up a couple of Suit Cases with soft Jack and then get out again, just like that.
1935		H.L. Davis Honey in the Horn 15: If you yank him out for any all-night sashay on these roads, you ought to be ashamed of yourself.
1947		W.D. Overholser Buckaroo’s Code (1948) 52: A smart alec, long-tongued, sashay-dude like you.