“Don’t let your alligator mouth overload your hummingbird ass” is a popular Texas colloquial phrase. “Don’t let your alligator mouth overload your mosquito asshole, hoss” was cited in print in 1975. “So you better just get Dick in here before your alligator mouth overloads your hummingbird ass” was cited in print in 1983.
 
“‘Don’t let your alligator mouth outstrip your killdee ass.’ (Also: ‘. . . hummingbird ass.’ )” was cited in the popular Texas slang book, Texas Crude: The How-to on Talkin’ Texan (1984).

 
Google Books
W.W. and the Dixie Dancekings
By Thomas Rickman
New York, NY: Pocket Books
1975
Pg. 45:
“Don’t let your alligator mouth overload your mosquito asshole, hoss.”
 
Google Books
The Franchise
By Peter Gent
New York, NY: Villard Books
1983
Pg. 73:
“You might be able to carry out your threats, but first you got to get out of this room. So you better just get Dick in here before your alligator mouth overloads your hummingbird ass.”
 
Google Books
Texas Crude:
The How-to on Talkin’ Texan
By Ken Weaver
New York, NY: E. P. Dutton
1984
Pg. 50:
“Don’t let your alligator mouth outstrip your killdee ass.” (Also: “. . . hummingbird ass.”)
 
22 October 1989, San Jose (CA) Mercury News, “Seifert perfects being practical,” pg. 1C:
’‘Maybe,” Seifert said, “it’s just that I remember the old saying: Don’t let your alligator mouth get you into situations that your canary butt can’t handle.”
(George Seifert, head coach of the San Francisco 49ers—ed.)
 
Google Books
Swan:
The Second Voyage
By Jim Moore
Dobbs Ferry, NY: Sheridan House
1994
Pg. 114:
Suddenly, I remembered an expression of a friend of mine: “Don’t let your alligator mouth outweigh your hummingbird ass.”
 
Google Books
Cassell’s Dictionary of Slang
By Jonathan Green
London: Cassell
2005
Pg. 428:
don’t let your (alligator) mouth overload your ass phr. [1960S+] (US) keep quiet, esp. in a difficult situation where words might complicate matters.
 
Google Books
The Complete Guide to Becoming a Successful Mortgage Broker:
Insider Secrets You Need to Know
By Patricia Hughes
Ocala, FL: Atlantic Publishing Group, Inc.
2008
Pg. 228:
Never let your alligator mouth overload your hummingbird ass. (I am from Texas; I thought you might appreciate a little colloquialism.)
   
Big Shirtless Ron
Thursday, March 26, 2009
Alligator mouth meets hummingbird ass
(...)
It goes back to a saying my father used to use on a fairly regular basis.  “Don’t let your alligator mouth overrule your hummingbird ass”.  Basically, if you are going to talk, be prepared to back it up.