OT: Florida climat question

Hi everybody, I have not been very present here lately, but still would like to have my South-East USA friends' advice :

We will auto-tour Florida and Louisiana during the first 2 weeks of June, first week Florida South ( Miami, the Keys, Orlando, etc..) then move on to New-Orleans, the Bayous, Memphis .....

No choice with dates, could not start earlier, so should we expect problems with bad climat, like hurricanes, as I just read (too late) that their official season starts 1st of june .........
Thanks
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JJ.