Right now we have major cities limiting or ending services like police responce and other "services" that were taken for granted.
As far as the interstate goes, I agree with Ranger, a big parking lot. I've seen numerous times when New Orleans has been evacuated. Their are use to doing it, with both sides of the interstate turned to out bound traffic. It still took 8 hours to go from NO to Baton Rouge, normally a 40 to 50 min drive.
In the beginning the state hwy and county roads will be the way to go. But that is not a given. During Katrina several towns surrounding NO blocked even foot traffic.
You might want to look at the DeLorme atlas & Gazetteer for every state you will have to cross. These books show even the smallest roads, that usually aren't of the state traffic maps, or road atlas.
Bottom line you are going to have to keep vigilant on current conditions. The first out will be the winners. One last point, FUEL!!! On all the evacuations listed ALL the gas stations ran out of fuel very quickly. Vehicles running out of gas on the road is one of the causes of the back ups. How much fuel does it take you to get to your location? Is it possable for you to carry that much fuel with you? At the very least you should keep at least 1 veh tank of fuel in reserve at all times.