We haven't stayed at the "Florida" Welcome center (going in) but we HAVE stayed at the Georgia Visitor Information Center (coming home) at Exit 1 on I95. We stayed for about 6 - 7 hours and had NO issues. We felt very comfortable and safe there. Also you can get to other facilities on that exit including a "Pilot Travel Center" which also has a lot of parking for the big rigs.
When traveling, I am a big fan of the Boondocking approach. I prefer to spend my time at the destination, so the traveling stops along the way are merely for "maintenance" (if you will). Resting, refueling vehicle (and family), and relieving.
I don't remember exactly where it was, but on one of our trips, we spent about 6 hours at a (smaller) rest stop about an hour from the Fort and somewhere on I4 I believe. That spot was pretty quiet and we pulled in about midnight - 1AM and slept until the morning. Again.. No issues for us.