It's all very well to say you're staying put in an emergency, but sometimes the situation requires a move. About 450 miles west of El Paso is the Palo Verde nuclear power station. It has three reactors, and is about 100 miles north of the border,in some of the most rugged border terrain you can find. If a bunch of terorists were to infiltrate from Mexico, and cause all the damping rods to be withrawn from the reactors, the radiation would be intense. I'm downwind, according to the prevailing winds, so if I stay put, I'll be exposed to intense radiation when the radioactive plume comes by. My choices are to move or to glow in the dark. I'm ready to move if I have to.
I think you seriously underestimate the security plans for these sites; if they are under threat, they have a set of protocols which go into effect automatically, and once the first step is initiated, the system becomes very, very difficult to tamper with and override.
As in- it's probably easier to get the codes for the nuke football from the POTUS than it is to turn a nuke power plant on its' head via violence.
With this said; again I will emphasize that fuel will be non-existent. Whatever you have is what you will have access to. To the end that were a really nasty hurricane or tornado wipe out your area, or say there was some kind of civil unrest...... you will not be able to move as you THINK you will. (I got mah truck, she's got 350hp and 650ft. lb of torque..... I'm gonna hitch the 5th wheel trailer to her and we are going to get out of here.....)

You will find roads blocked by nature. Blocked by blockheads out of fuel. Blocked by LEO/ National Guard.
I hate to say this, but people REALLY need to learn to travel on foot. You, the missus, and the kids need to take a nice, long walk in a given general direction, to a friends' house with nothing more than a couple bottles of water and a couple snacks in a backpack. Find out just how long it takes to cover say a measured 2 miles. 5 would be better. With the crew in tow, and no extra weight, you will quickly realize just how much more important it is to look to your surroundings and really figure what is possible close to home. It's slow, slow going when packing gear, a single firearm, food and some clothes..... especially when the kids get tired and you are now humping along with a rugrat on your shoulders.
Trust me; try it, just once. You will seriously look at your plans of leaving differently. You will alter your ideas of destination, quickly. You will have to figure what will be left behind, what will be taken along, what will be burned, and what will be buried for later retrieval.