Unusual looking twig with some side axillary buds. Similar tree elsewhere had more normal looking buds. I believe this could be Q. laurifolia. Leaves are persistent as shown in the photos. According to the Trees of Northern Florida, this species is evergreen and Q. laurifolia is "tardily deciduous".
Distinctive bark and diagnosis of the twigs leads me to believe this is C. laevigata. Note that is has a chambered pith in the area near the leaf nodes.
In full flower just as the leaves are starting to emerge. Difficult to distinguish from P. umbellata but the form (thicket instead of single trunk) leads me to this. After leaves are fully formed I can check for glandular tips.
This oak species is only known in North Florida and coastal Georgia and South Carolina. Often confused with Q. alba but differs by shallower lobes and narrower sinuses. Q. austrina is much more common in the Tallahassee area than Q. alba and is probably confused with that species. It has similar looking bark.