Part of my bug out kit includes the family tent, it is a 8 person tent that is hanging in the garage next to the vehicles. Not that I don't trust .gov but I don't trust .gov! I'll take my changes on my own with my family. Katrina is an example of .gov shelters gone bad in an area, but you have to also look at the "clients" that were going to the shelters in the city area, or at least the ones that made the news, these were people that really didn't want to help themselves for one reason or another, also many criminal type people. I find it hard to believe that when you have days warning to evacuate an area you have a problem getting out, tells me you don't want to leave. KC5FM makes a good point that in other area's the shelters were set up and ready to accept and help anyone needing it. You have to make the decision before a disaster of what your course of action will be, can I get out by vehicle? Do I sit tight and wait it out? Do I have the supplies to last? FEMA suggest a 72 hour kit, I suggest much more, I want to be able to last weeks/months on my supplies if I decide to stick it out at my current location, but my current location is different than yours for the most part I can wait something out, for me snow is the biggest disaster we have to deal with, tornado's every now and then but nothing near what the gulf cost people deal with when a hurricane comes ashore
If you visit this site or sites like these you are either prepared for disasters or are in the works of being prepared for disasters, you are on the right track and if you continue you will find that in a disaster you don't have to rely on a shelter, you will have the means to get by on your own in comfort with everything you need even sleeping in a vehicle, until you can return to normal life.
If you haven't camped I would suggest learning the skills and having some equipment, you can set a tent any month of the year, they make heaters for them for people like me in the cold climates. I have camping cook stoves both for backpacking (jetboil type) and larger 2 burner stoves with fuel, dutch ovens for making stews and baking. The car, or in my case the car and van, would become a harder shelter to go to if conditions around me got bad and I didn't feel comfortable in a tent.
The last thing to think about is the cause that would make you want to leave, is it natural like storms? is it man made, terrorist attack? For me each has a different reaction, a bad storm for me would not be so wide spread that it would destroy the area, again snow is my biggest issue, for that I sit tight, a tornado is also pretty localized so family across town would be my course of action, a terrorist attack is a completely different animal, although we hope that something is local in nature it might be local to a city, for example the things going on in India over the past couple of days, 9/11 in our country, which took place in 2 cities and brought our air transportation to a standstill, I had people working in NYC during 9/11 and also spread out across the country, I spent the day trying to get my crew out of New York, realizing that they didn't have any supplies with them, not even a good map, cell phones were spotty and hotels in the city were not going to be an option as we didn't know how things were going to unfold during that day, the best action was to bug out and do it as quickly as possible. I spent the day on the phone with them and a map up on my computer giving them directions until they got far enough out of the city to then be on there own safely.
No one plans on sleeping in the car, but you better have that plan in your head or when it happens the stress of that situation can be overwhelming and limit your ability to get through the situation at hand.
Sorry for rambling on here