As I understand it your car is the same as your house in Georgia.  No worries.

You're doubly fine if you have a Florida carry permit, but perfectly fine without it. I type this while sitting in a hotel room in New Jersey, where it is probably illegal for me to possess a nerf gun.

ETA safe travels for you and the family

You're fine with regards to Georgia law.

However, there's a "gotcha" with Federal law.  The Gun-Free School Zones Act defines a "school zone" as being within 1,000 feet of a school.  There's plenty to know about the GFSZA, but relevant to your question, the NRA-ILA says this:

Federal law doesn't consider states having reciprocity to mean you're "licensed" in another state -- it only applies in your state.  So, even if you have a license with reciprocity with Georgia, when you're in Georgia you must have them unloaded and locked in a container according to Federal law.  I've also seen analysis that says if they're unloaded with the ammunition separated (and both not in the same compartment as any occupant), they do not have to be locked (but I ain't a lawyer).

Good luck!

I wouldn't worry about this, I doubt you'll find FBI agents pulling people over in school zones to look for guns, and this particular law has been thrown out by the US Supreme Court once already, all Congress did was throw in some verbiage claiming that schools are important to interstate commerce and passed the same law again, and it was all prior to _Heller_ and _McDonald_.

Georgia is more gun-friendly than Florida, open carry is legal with a carry permit/weapons license, there are relatively few places legally off-limits to carry (courthouses, government buildings with security screening, churches, jails, mental hospitals, and secure areas of airports, etc), signs have no particular legal weight, etc...

I agree it's not something to be worried about and that there aren't going to be FBI agents trolling for victims, but he did ask what the law was.