Hi there!!
We drove from Houston to Destin, FL the middle of September (about 3 weeks after Katrina). There was a bridge out on the border of MS and AL, but that should be fine by now. We had filled our tank before Baton Rouge (Lafayette, LA), then did not need gas until we were in Destin. The roads for us were very full (with cars, but also lots of debris) on I-12. It should be much better now. Our friends just came back from WDW and got a hotel just after Mobile, AL which is about 7 hours from Houston. Most everything after Mobile had rooms. We did not have trouble finding gas. I would think that if we did not have trouble with gas 3 weeks after the hurricane, you all should be fine with the gas supply.
I think our family are suckers for punishment, we drive straight through to WDW from Houston--15 long hours, but then a nice long time to relax at WDW!!
Good luck with all of your trip planning!!