Drive safely and rest well.
It is both illegal and extremely dangerous to stop for 6 hours at a rest area, unless you have permission from the security guard. In FL they have had some bad experiences with people who chose to do just that., including the murder of a family of four tourists, some years ago. Since then many of the rest areas have had armed security, but unfortunately they are all along the east coast. There are only two or three truck stops which will allow overnight parking and use of their facilities. One is south of Jacksonville, the other is just south of Kissimee. (I have stayed at both, my vehicle is set up with a bed etc. for a comfortable's night sleep. Curtains on the windows and plugs in ears.)
But you will need to stop before you get to FL. Divide the miles in half and look for a cheap motel in that area. Use a coupon booklet or coupon off the internet to get motel discounts.
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I have never road tripped like this, ....
That is obvious. If you had you would not make the suggestion of doing this trip without a proper night's sleep. If you think that you will be able to get any sleep at all sitting up in a car, in the noise and light of a busy truck stop or rest area, you are deluding yourself. You may be able to doze for a few moments, but you will not be in any condition to be a safe driver. Two of you with the doors locked and the windows shut will soon become very uncomfortable. With the windows open you won't be able to sleep for worry of been broken into. Studies have shown what Michael stated. Fatigue is the insiduous enemy of driving long distance, mainly because you are not aware of it coming on, and by the time you are tired, you will have been driving for a long time with reduced concentration, reaction, and all the other matters which are needed to drive safely. It is the cause of thousands of deaths on US roads every month.
Lifey