I don't know why you would leave FL for CA, but that's your biz. I assume you are traveling I-10, which I have done several dozen times in my WK2.
There are no rest stops I can find on the highway in Louisiana. Places to avoid during morning / evening rush hour: Baton Rouge, Houston, & San Antonio. If you are planning to go through NOLA, avoid it at rush hour, too.
You might consider getting a CB to talk with the truckers about road conditions ahead. I have a hand-held, but can't leave it on due to the foul language. I mean, I was in the Navy, but these guys are just foul.
Remember, you are driving a Jeep. The median can be used to turn around. Just be on the lookout for fences and grass covered concrete drain covers. And LEOs.
Make sure you have phone with a map program on it. It really can navigate you around a tie-up. You might also get a holder for it, so you can see it without picking it up. Make sure you have the right charger that works with your Jeep.
Take plenty of water with you and stay hydrated. You are contemplating a long-ass trip. Might also want to take some food. I usually take pretzels and Snickers.
Once west of SA, biggest problem I found is boredom. There is just nothing there but highway and dirt. We went to Los Alamos a few years ago and I felt like I was driving on the moon in New Mexico.
Make sure you have a decent spare. Get a good lug nut wrench and a jack. I now carry a few foot long 2 x 10s with a bottle jack. Waaayyy better that that little toy jack that came with your Jeep. We have AAA, but it isn't any good anymore. If you rely on them, count on a minimum of 2 hour wait.
Finally, you can't do this trip non-stop. Sit down with the map and plan your time. Figure an AVERAGE speed of around 60 or 65 mph to see where you will be at what time. Make motel reservations before you leave! Last trip, I just figured we would stop when we got tired, but could not find a room in any motel on the interstate for an hour.
One last point. I like to drive at night, but I am a night person. I can drive from 7 p.m. to 7 a.m. with no problem. There is a lot less traffic on the highway at night, but the view isn't very good. Think about when you want to drive. I would estimate about 5 mph added to the average if you drive at night. There are plenty of gas stations along the route that are open all night. If you don't have one, get a credit card for gas so you can use the automatic pumps at 3 in the a.m.
One word of warning: when you get to Texas be careful. You may just want to stop and stay here!
Be safe.
