I asked what the Historic New Orleans Collection had, and I was kindly given a page from the 1964 book below. The song "Begone, Dull Care" was sung at Mardi Gras in the late 1800s, but the nickname does not date to the 19th century.
Author Reinders, Robert C.
Title End of an era: New Orleans, 1850-1860.
Imprint New Orleans, Pelican Pub. Co. 
One hundred years ago, as today, New Orleans was billed as "the city care forgot." A combination of forces - French traditions, sea port town, frontier influences, wealth - gave New Orleans in the 1850's a reputation as the most glamorous, and most decadent, city in America.
18 December 1910, Beaumont (TX) Enterprise and Journal, pg. 4 ad:
"THE CITY CARE FORGOT."
America's COnvention and Carnival City.
ST. CHARLES HOTEL
9 February 1912, The Daily Picayune (New Orleans, Louisiana), third section, pg. 13, cols. 4-7 ad:
"THE CITY CARE FORGOT"
The St. Charles
"The Center of the City's Hotel Life."
21 April 1913, Sheboygan (WI) Press, pg. 6, col. 5:
"The more the merrier" will be the slogan Convention week, and Baltimore will be "The City Care Forgot."
(Baltimore? The article mentions "merry masqueraders, a la the Mardi-Gras in New Orleans" -- ed.)
14 June 1914, Washington Post, pg. 5, col. 6:
You will remember that when Morello was in New Orleans on the affairs of the Ignatz Florio Association he wore a bandanna handkerchief with five knots toes in it in defiance of those who threatened to epose his counterfeiting schemes in connection with the real estate propaganda. All afternoon Morello stalked the streets of "The City of Care Forgot" wearing this badge of defiance
described in the Black Hand ritual as worn by the head of crime.
7 February 1921, Atlanta Constitution, pg. 6?, col. 7:
On the days of its prime carnival season saw its official opening the Monday preceding Shrove Tuesday (mardi gras) when with due formality committees of the Mystic Krewes sailed down the river to meet the royal yacht, aboard which was Rex, King of Misrule, and for two days sovereign of the so-called City that Care Forgot.
17 February 1931, Chronicle-Telegram(Elyria, Ohio), pg. 10?, col. 5:
... -- Mardi Gras. All New Orleans seemed to forget business and the city seemed to be the one "care forgot."
Nicknames of Other Places • Big Easy, City That Care Forgot (New Orleans nicknames) • (0) Comments • Tuesday, March 01, 2005 • Permalink