Wow, just looked out of curiosity, and I'd give Allegiant a shot for that kind of a deal. I looked in Nov. when I'll be going back for work and they had round trip, non-stop from AUS to VPS for $101! Crazy cheap, wish they flew out of DFW/DAL.
Southwest flies into ECP and PNS, but they they stop in DAL and HOU, respectively. Depending on where you're staying they're both roughly an hour from Destin (ECP closer to east side/Sandestin/30a and PNS closer to west side/Ft. Walton/Okaloosa Island).
We've made the trip to/from DFW about twice every year for 10+ years. When we lived there, we had my wife's folks in DFW to visit, and now we're in DFW and have my folks to visit in the Destin area (Niceville). Our kids are 10, 4, and 2 now, but they've all made the trip all their lives and they all do great. Of course, we suspend some rules to keep them happy, so unlimited screen time and an abundance of junk food certainly helps. Usually 12 hours is a pretty good estimate.
I don't go quite as early as planoaggie, but we try to get on the road by 4-430. Definitely helps if you can boogie the first few hours whith minimal traffic/stops.