“Sometimes I sits and thinks, and sometimes I just sits” is a jocular saying about an old person, but has been used to describe any person of inaction (such as someone in government). The saying was printed in several American newspapers in March 1905:
 
A bond salesman just back from Maine said he asked an old fisherman in a snowbound hamlet what he did with himself evenings. 
The reply was: “Oh, sometimes I sit and think, and sometimes I just sit.”—Boston Record.
 
“Sometimes I sits and thinks, and then again I just sits” appeared in the English humor magazine Punch on October 24, 1906. “Sometimes I sits and thinks, and sometimes I just sits” is a popular form of the saying.
 
“A conservative man is a man who just sits and thinks, mostly sits” is a related saying.
 
[This saying was later researched by the Quote Investigator.}
 
   
18 February 1905, Pittsburg (PA) Press pg. 2, col. 5:
How He Spent His Time.
A bond salesman just back from Maine said he asked an old fisherman in a snowbound hamlet what he did with himself evenings.
 
The reply was: “Oh, sometimes I sit and think, and sometimes I just sit.”—Boston Record.
 
7 March 1905, Trenton (NJ) Times, “Snap Shots of Humor,” pg. 5, col. 4:
THE SIMPLE LIFE.
A bond salesman just back from Maine said he asked an old fisherman in a snowbound hamlet what he did with himself evenings.
 
The reply was: “Oh, sometimes I sit and think, and sometimes I just sit.”—Boston Record.
 
Chronicling America
19 March 1905, Los Angeles (CA) Herald, “Smiles,” pg. 6m col. 5:
A bond salesman just back from Maine said he asked an old fisherman in a snowbound hamlet what he did with himself evenings.
 
The reply was: “Oh, sometimes I sit and think, and sometimes I just sit.”—Boston Record.
 
22 November 1906, Philadephia (PA) Inquirer, pg. 8, col. 4:
Variety of Diversions.
Vicar’s wife (sympathetically)—Now that you can’t get about, and are not able to read, how do you manage to occupy the time?
Old man—Well, mum, sometimes I sits and thinks; and then again I just sits.
—Punch.
 
22 April 1911, Boston (MA) Herald, “Appreciations of New Books,” pg 8, col. 3:
He thus describes life on board—albeit the description does not answer all the exigencies of the cruise: “Wonderful how smoothly the time passes, and yet it is not very clear how it is filled. There is some reading done, a deal of sleeping and much eating, but for the most part it rather resembles the days of a bucolic old gentleman who was asked, since he was past bucolic activities, what he did with his time. ‘Sometimes,’ he said, ‘I sits and thinks, and at other times I just sits.’
 
Google Books
“Life is celebration not a search for success”:
Studies in the Poetry of Peter Russell
By Anthony L. Johnson
Salzburg: Institut für Anglistik und Amerikanistik, Universität Salzburg
1995
Pg. 72:
The phrase “he sits and thinks” is another cultural allusion; it takes up a gag used by American comedians in the 1950’s, “Sometimes he sits and thinks, and sometimes he just sits”.
 
Google Books
My Mama Always Said…:
A Book of Southernisms
Collected by Marillyn Taylor Klam
Illustrated by Allyne Petty
Gretna, LA: Pelican Publishing Company, Inc.
1996
Pg. 94:
Another quote from Mammy was
 
Sometimes I sits and thinks.
Sometimes I just sits.
 
Google Books
Train Your Brain:
Teach Yourself
By Simon Wootton and Terry Horne
London: Teach Yourself
2010
Pg. ?:
‘Sometimes I just sits and thinks, and sometimes I just sits.’ (Mark Twain)
 
Google Books
Concise Oxford Dictionary of Quotations

Edited by Susan Ratcliffe
New York, NY: Oxford University Press
2011
Pg. 300:
Punch 1841-1992
English humorous weekly periodical
Sometimes I sits and thinks, and then again I just sits.
24 October 1906