That's not how that works, syntactically or logically. The sentence has a distinct meaning that is not preserved when trying to cancel our the negations like that. (This isn't "I don't owe nobody nothing".)
Less concisely, it means that it does prevent the government from telling you you can't move around, but it doesn't prevent the government from forbidding you to drive.
It absolutely does not mean that they can't tell you you're not allowed to drive. Driving is, and always has been a privilege in the US; not a right.