Pick your season, plan accordingly.
If you like hot, you can avoid the high season winter months. If you would like to warm your bones for the winter, no better place.
No matter when you decide to go, the best value is the state parks, so they book early all year.
If you go in the summer, availability at most CGs is much better.
First thing out, you should try to reserve at any of the state parks or skip from one to another. Reservations are difficult, but you may get lucky and string a week or two together with some skipping around. The SPs book up mostly on the first day allowed to reserve, so get up early and log in right at the reservable limit. I think its 11 months out, maybe 10 -forget.
Bahia Honda will be the toughest. The others are a little better. The Class B will open up more sites for you too.
For instance, I am planning a 2-week trip June-July timeframe right now. Bahia Honda only has two nights available. They are non-consecutive at different sites for the period I am searching. The rest are reserved for that two-week period. We may snag one night to be able to stay over and enjoy the best beach in the Keys (Sandspur Beach) for two days. The other SPs in the Keys have many more sites available for that timeframe.
We are lucky to have the Navy Base CG available to us. So, we can come and go there as we please during that time of year.
Since you mentioned history, check out a possible trip to the Dry Tortugas and Fort Jefferson. That should make for a great day trip. We are planning that for the first time this trip.
Since you don't like the nightlife, you could take in the town of Key West in a day if you'd like. The tourist stops are best done on the trolley the first time down. Then, take it from there.