I believe the origin of the word "Dixie" is "dix" in French, referring to 
a ten franc note in Louisiana, but I'm interested in the origin of the 
expression "took a dixie" meaning a "fall" or "disappearance", as in 
"taking a powder"

I wonder if it's akin to "went south?"

I think Nellie_Bly is probably correct. 

"To go south 'vanish, abscond,' is 1920s, Amer.Eng., probably from
mid-19c. notion of disappearing south to Mexico or Texas to escape
pursuit or responsibility, reinforced by Native American belief
(attested in colonial writing mid-18c.) that the soul journeys south
after death."

http://www.geocities.com/etymonline/s9etym.htm

The etymology of "dixie" is sometimes related to the Mason-Dixon line.
There's an interesting discussion of "dixie's" origins here:

http://www2.h-net.msu.edu/~south/archives/threads/dixie.html

The phrase "took a dixie" or "take a dixie" is not in common usage in
any region of the United States.  If you have heard it used, it is a
locally coined phrase.  It is also possible that you misheard "take a
dive" for "take a dixie".  I hope this helps.