The commercial database (at large academic libraries) LION takes this quote back to 1822, in C. S. (Charles S.) Talbot () Paddy's Trip to America; or, The Husband with Three Wives. A Farce, in Two Acts New-York Printed for the Author 1822 48 p. Text type: Prose
Genre: Farce, Comedy First performed: Washington Hall, New York
First performed: 1821 or 1822.
The exact quote is: O'Fla. [Patrick O'Flaherty] [39] Can't your honour; I'm as dry as a whistle, [40] and could'nt squeese a word out, 'till I be after [41] getting something to drink.
JSTOR gets zero hits for the phrase "dry as a whistle", no doubt because "as" and "a" are stop words. The search: dry and whistle gets 1522 hits, and when you sort by "oldest to most recent" you get 3 hits older that 1839. But they all seem to have only one of the words on "page of first match".