Hi, @PalmWarbler. I have mostly birded the panhandle/Big Bend, central Florida, and the Tampa Bay area, so don't have personal experience with birding the east coast.
1. Panhandle/Big Bend (W to E):
St. Marks NWR (as has already been mentioned); we like to stay at the lodge at Wakulla Springs SP (some birding there, also). They give a discount to Audubon members. There are no TVs in the rooms, which is a ++++ for us
St. Joseph Peninsula & Apalachicola area--off the beaten path (although of late it has been, unfortunately, discovered) but well worth it, both for birds and a taste of the real Florida.
2. Central FL:
Sweetwater Wetlands (https://www.sweetwaterwetlands.org/)
Payne's Prairie (https://www.floridastateparks.org/parks-and-trails/paynes-prairie-preserve-state-park)
3. Tampa Bay area:
Ft. Desoto (https://www.pinellascounty.org/park/05_Ft_DeSoto.htm#general)
Boyd Hill Nature Park (https://www.stpeteparksrec.org/boydhillpreserve/)
The Celery Fields (https://www.sarasotaaudubon.org/online-guide-favorite-local-birding-hotspots/#1568384830680-d03a2fe4-fc8a)
4. South Florida-- @Connor Cochrane can help with suggestions for this area
Audubon Corkscrew Swamp Sanctuary (https://corkscrew.audubon.org/)
Myakka River SP (https://www.floridastateparks.org/parks-and-trails/myakka-river-state-park)
Dry Tortugas/Key West We have our first trip planned for next Spring, so I can't yet give you any personal reviews, but, by all accounts, a place all birders should eventually go. Only accessible by boat.
The beautiful thing about birding in Florida is that it can be a passive activity. There are birds everywhere. You will see flocks of White Ibis foraging in yards, Great Egrets eating snakes on house rooftops, Roseate Spoonbills caually foraging in ditches by the roadside, and Sandhill Cranes from the comfort of your car as you speed down the interstate. Have fun!