There is some useful shared cultural vocabulary, including "paper", "to write", "lime", "maize (corn)", "leached corn", and "to shell corn", and it looks like Caddo - spoken just upriver - in turn borrowed much of its maize-related vocabulary from Chitimacha. In combination with archeological evidence, this leads the authors to favour a migration date either some time around 850 AD, when the Caddo began low-level maize cultivation, or sometime around 1200-1450 AD, when they intensified it. Such a late date seems a little troubling, given how few cognates are to be found; Korandje separated from Songhay around 1200 AD, and there are well over 200 shared items there, mostly belonging to basic vocabulary. The ancestor of Chitimacha would have to have already been rather different from any other Totozoquean language even before they reached Louisiana; but then why did they apparently leave no trace in Mexico itself? Perhaps a study of southern Mexican place names could shed some light on the question.
This looks like historical linguistics at its best: a surprising long-distance connection affecting both language and culture. Now it's up to the historians and archeologists to fill in the gaps: why did southern Mexicans find it worth while to cross the Gulf to Louisiana in significant numbers?