It's not looking too good for anything connected to the Suwannee, Withlacoochee or Santa Fe Rivers. I just checked and the following State Parks are closed due to brown-out/flooding: Troy, Peacock, Madison Blue, Lafayette Blue and Fanning (even though Fanning's more swim/snorkel than dive). Also, Ichetucknee has an alert for extra long float times due to the recent rise on the Santa Fe which is still going up. Little River is a county park vs. State park but I'll bet it's blown out as well since it is close to Troy. The important river gauges to watch for the parks noted above are:
Madison Blue State Park
Withlacoochee River at Madison Blue SP
1 week ago: 43.42ft Today: 59.06ft (Needs to be in the low-mid 30s)
Peacock Springs State Park
Suwannee River at Luraville
1 week ago: 27.91 Today: 32.49 (Peacock closed on Tuesday 8/27)
Lafayette Blue Springs State Park
Troy Springs State Park
Little River
Suwannee River at Troy Springs Pool
1 week ago: 21.98 Today: 26.09
Fanning Springs State Park
Suwannee River at Wilcox
1 week ago: 6.98 Today: 8.32
As for Ginnie, I'd call ahead before heading up there. I also heard somewhere a while back that when the Santa Fe gets high they put some kind of silt fence material out across entrance to the Devil's run at its junction with the river to help keep brown water from getting too far into the run. I've never seen this personally but you can ask about it if you contact them.
The other well-known spots that are either isolated or not subject to flooding (Devils Den, Blue Grotto, Paradise Spring, Alexander Springs, Rainbow River, Catfish Hotel (at Manatee Springs) & Blue Springs (Volusia)) should be OK but will probably be very crowded. (Plus DD, BG & Paradise are pricey to enter.) If you want to try somewhere that's off the beaten path or not quite as popular there's always Hudson Grotto, Crystal River (King's Spring) or even Hospital Hole in the Weeki Wachee River.