There are so many tourists and snowbirds in FL that they aren't pulling people over for out of state plates. So no, you probably aren't going to be taken away in cuffs. There are fines though.
You're required to register your vehicle within ten days of becoming a resident. I think you could buy yourself a few more days if you sign up for an appointment within that ten day window but book the appointment out into the future another week or two. If you wait too long though, they'll charge you delinquent fees when you eventually do register for your plates.
Also, don't bother trying to save some money by letting your current tags expire. They'll back-date your tags to expire based on your previous expiration date.
I think the deadline for your drivers license is 30 days, but you might as well deal with that at the same time you go in for your plates.
The requirements are set by the state, but the actual work is done by the counties. So you'll need to go to the tax collection agency for your county to make the appointment.
Here's a link to the state's New Resident page:
https://www.flhsmv.gov/new-resident/