When the weather gets bad, the airport brings out Coleman (army-style) cots for distressed passengers. I don;t know if they are available on normal overnights.
Failing that, if you don't want to sleep on the floor, the best bet may be the "DFW Airport Travel Lounges" (formerly Samsung Lounges), which are small, free spaces located at gates A29, B8, B28, C8, C27, E8, E31, where you'll find the cushiest chairs at DFW.