"The only thing I've ever really thought about is the idea of getting into an accident on my way to Austin to claim my prize. ... knowing that I'd be driving with (essentially) millions in the car would add some paranoia."
Anyone who can run around with millions of dollars in their pocket and not be paranoid probably doesn't understand their situation. As I recall, when the ten (?) machinists in Ohio won about $300 million in '98 two of them were already on the way to Indiana when another contacted a lawyer. The lawyer was able to call them back, and had the ticket deliverred by armored car. That might be paranoid, but it doesn't seem excessively paranoid under the circumstances.
We see a lot of posts by people who say they'd be waiitng when the lottery opened the doors on the first business day after they won, but you can't do that and also take care of a lot of the things that are good ideas for somebody suddenly worth millions.
I'll assume your worry is about being hurt in an accident, but consider this scenario. You don't get a scratch, but the vehicle you sideswiped hits a bus with 20 school kids, and several adult chaperones, some of them high earners, because it's a field trip. As a purely financial matter, dead kids aren't usually worth all that much. OTOH, crippled kids and parents who can no longer support their family are worth an awful lot. Sure, it's very unlikely, but you're making that trip because you've got an ability to beat steep odds. If that happens you should probably give the ticket to a friend or relative who will hopefuly help you out later on, because all that money is about to go to other people.
Not that you can't have an accident really close to home, but my first trips would be to the insurance agent and the lawyer that specializes in financial matters, to make sure that what's mine stays mine. I also don't want to have to move directly into a motel, or even a fancy resort. If it's a substantial prize I don't want to be hard to find for a week or a month. I want to be hard to find for the rest of my life, and I'm never coming back to the former residence. I've got to talk to people who can delete references to me on a few websites, and wait for google to forget I exist. Assuming I have to appear at a press conference I need time for my hair and beard to grow until I look like Joaquin Phoenix. Rushing probably won't result in an accident, but there are plenty of good reason to sit back and wait a while.