To spread oneself too thin doesn't have an entry in the OED, but goes back to the late 19th century.
The oldest I found is from the 1893 Report of the Twenty-Third Annual Meeting of the Vermont Dairymen's Association:
Why, when a man undertakes to spread himself over the whole work embraced in agriculture, even if he is a pretty smart man, he will spread himself too thin. (Laughter.) No man should attempt so big an undertaking.
Here's a "spread himself too thin" in an 1894 newspaper (Mexico Weekly Ledger, September 06, 1894):
Gov. McKinley is going to speak
in the Maine campaign. If he doesn't
mind he will spread himself too thin
to hold that nomination.
The exact phrase "spread myself too thin" goes back to at least 1942 in The Saddle and the Plow: An Historical Novel of Texas by Ross McLaury Taylor (page 167, confirmation):
"I thought everything was all right with you!"
"Yeah! Me, too. Reckon I spread myself too thin." Fielding massaged the bulbous tip of his nose with a bent forefinger and thumb.
Another is from Billboard magazine of 14 August 1948 ("Early Success Reflects Capable Act Merchandising by Al Martin"):
"I haven't spread myself too thin," Al says. "As a result of this I have been forced, for the time being, to pass up some good business that has been offered to me in the Middle West."
US v. UK
This ngram chart shows, of the texts indexed by Google Books, it's more common in US English than British English.