“Busier than a one-armed paper hanger” is one way to say that someone is very busy. “Our chef is kept as busy as a one-armed man engaged in the exercise of bean-bag” was printed in The Pacific Commercial Advertiser (Honolulu, HI) on June 19, 1906. “And then I got as busy as a one-armed man with the nettle-rash pasting on wall-paper” was written by O. Henry (1862-1910) in the short story “The Ethics of Pig” in the October 1906 Munsey’s Magazine.
 
“A man with red grogans was the chef and he was as busy as a one-armed paper hanger with the hives” was printed in the Buffalo (NY) Enquirer on October 15, 1906, written by “Tad” Dorgan (1877-1929). The saying is now infrequently used and is regarded as politically incorrect.
 
“Busier than a one-legged man in an ass kicking contest” is a similar saying.
 
       
Chronicling America
19 June 1906, The Pacific Commercial Advertiser (Honolulu, HI), pg. 5, col. 4:
THE FAVORITE GROTTO.
Our chef is kept as busy as a one-armed man engaged in the exercise of bean-bag.
 
The Unz Review
October 1906, Munsey’s Magazine, “The Ethics of Pig” by O. Henry, pg. 66, col. 2:
And then I got as busy as a one-armed man with the nettle-rash pasting on wall-paper.
 
Newspapers.com
15 October 1906, Buffalo (NY) Enquirer, “Sunday Prizefight in Little Old New York” by Tad, pg. 8, col. 1:
A man with red grogans was the chef and he was as busy as a one-armed paper hanger with the hives.
     
Newspapers.com
20 October 1906, The Evening Star (Washington, DC), “Novices at Kicking,” pt. 3, pg. 2, col. 7:
The next man up at the cashier’s window was a Russian grand duke, a young man said to be worth some hundreds of millions of dollars, a spender from away back, a man who on a tour of this country was as busy as a one-armed paper-hanger with the hives just tossing away money. 
(...)
CLARENCE L. CULLEN
 
28 April 1907, Philadelphia (PA) Inquirer, “Letters from an old sport to his son at college” by Jim Nasium, third sec., pg. 6, col. 2:
If you would succeed in base ball when you get on the field you want to keep busier than a one-armed paperhanger with the hives.
 
5 May 1907, The Sunday News (Charleston, SC), “Base Ball Fame Hard to Attain,” pg. 17, col. 1:
If you would succeed in base ball when you get on the field you want to keep busier than a one-armed paper hanger with the hives.
 
24 June 1907, The News and Courier (Charleston, SC), “Advice To Young Ball Players,” pg. 6, col. 5:
So you stick right on the job, and keep busier than a one-armed bill poster with the itch, working in a high wind.
 
Chronicling America
26 June 1908, The Arizona Republican (Phoenix, AZ), pg. 6 col. 4:
The restaurants are preparing for the serving of an unusually large crowd and one is likely to e busier than a one-armed man with the seven-year itch.
 
Chronicling America
8 June 1910, The Salt Lake Tribune (Salt Lake City, UT), “‘Strictly Fresh’ Eggs Down to Thirty-five,” pg. 17, col. 4:
The vender who offered the stupendous “bargain” was busier than a one-armed billposter in a gale of wind for a few minutes supplying customers, who fairly fought with each other to get to the counter.
 
Google Books
January 1921, Boys’ Life, “Not in the Curriculum”  by G. G. L., pg. 12, col. 1:
It looks as though you were going to be about as busy as the Prince of Wales was when he was visiting in New York, and he was busier than a one armed paper hanger with St. Vitus dance.
 
Google Books
1 October 1921, The Saturday Evening Post, pg. 19, col. 1:
Seriously I wish you success with your next song and I laughed when I read your remark about being busier than a one armed paper hanger. I don’t see how you think up all those comparisons and crazy things to say.
 
Urban Dictionary
busier than a one armed paper hanger
To be so busy, it will be impossible to finish your task.
Bob: “Hey pal, can you give me a hand with this task.”
You: “Are you serious?” “Im busier than a one armed paper hanger.”
by Frainslug November 04, 2005
 
Google Books
In a Manner of Speaking:
Phrases, Expressions, and Proverbs and How We Use and Misuse THem
By Colin McNairn
New York, NY: Skyhorse Publishing, Inc.
2015
Pg. ?:
Some absurd images carry an overtone of mockery. When we describe someone as “busier than a one-armed paper-hanger” or “busier than a one-legged man at an ass kicking contest,” we risk giving offence. Because they draw attention to the limitations resulting from certain physical disabilities, these expressions are sometimes viewed as politically incorrect and, therefore, to be avoided.