I live in FL and yes, the turnpike rest areas have everything you need including security... and the other rest areas on the main roads usually do... the problem is that if they know you are sleeping in your car, they will wake you and have you move on
A short nap, they will probably allow, to keep you safe from falling asleep while driving.
BUT I slept in my car at rest stops long before it wasn't allowed.... and no, it isn't safe. (Talk about a reason to carry a firearm?)
I am an experienced camper... tent and otherwise... and there is always the risk of someone aggressive coming along... and it results (that risk) in not sleeping soundly.
No sleep for 24 hours (or inadequate non restorative sleep) results in the same effects as having .10 blood alcohol, btw.
Camping in State Parks is the way to go
BTW.. wanting to get away is normal for traumatized or otherwise overwhelmed ppl. I often want to just get into my car and drive "North."
But my T talks me out of it...besides, wherever I go there I am anyway. If it's situations you need to clear your head from, then try shorter excursions doing things that take your mind totally off your situation. (I do very short: movies, medium short: zoo, parrot jungle, mall window shopping, and not short: Disneyworld.) oooo camping at Disneyworld is great fun btw, and they have horseback riding
which I can't do anymore
Ok I've rambled long enough. Staying in your car in service plazas or rest areas is NOT safe. Do so only if driving while asleep is more hazardous to you.
((((hugs)))) stay safe!