Do you have a drivers licence? Real ID or otherwise?
Or would you be traveling on your passport?
Either way, you're technically authorized to stay in the US legally, and you should technically be totally fine even if asked. You should always have proof of the change of status on you, assuming you're changing status.
If you're driving to FL, there's a lesser chance of being asked--you'd have to be pulled over, and if you have a valid driver's license, it's unlikely to be questioned. If it's questioned, you're still OK, but you never know how knowledgeable police officers are about these things.
If you're flying, they ask for a valid US government issued ID. If you have a drivers license, even if it's not real ID (until 2020), TSA won't check anything but. They won't even question your passport. However, the airport might have ICE agents who do check people's documents, and again, they might question you--but you have valid documentation. It just depends on how familiar they are with it.
So all in all, you can move within the country freely, by car, plane, train, or otherwise, but you may have to answer more questions than necessary, and potentially be detained to answer those questions. You should have paperwork with you. The chance of this is small, but not zero. In the end, worst case scenario, you'll still be OK.