I offered some info about the recommended Google Maps route between Columbia, TN and Mexico earlier.
Here is some similar info related to the route between Columbia, TN and New Orleans, a place where Tad helped during Hurricane Relief efforts.
Click to enlarge map.
There is a state park, Bogue Chiito State Park, along the route that does offer camping:
http://crt.louisiana.gov/louisiana-state-parks/parks/bogue-chitto-state-park/index
Wooded areas with public access, but no advertised camping:
--Bayou Sauvage: Wildlife Refuge:
https://www.fws.gov/refuge/Bayou_Sauvage/about.html
--Salvador Wildlife Area:
http://www.wlf.louisiana.gov/wma/2765
--Maurepas Swamp:
http://www.wlf.louisiana.gov/wma/2791
--Biloxi State Wildlife:
http://www.wlf.louisiana.gov/wma/32647
In New Orleans, there were 9 AirBnB locations offering a room for under $20 a night:
https://www.airbnb.com/s/new-orlean...price_max=20&allow_override[]=&s_tag=pVdZzPU_
There are many, many more AirBnB options if you go up a few dollars.
Here, someone describes a somewhat "off the grid" lifestyle on the bayou:
http://culturecurious.net/life-on-the-bayou/