Germany

I love September, particularly the second half, the weather is nice - warm days and slightly cool nights - and the crowds have settled down. It's harvest season and there are many fests scheduled in September (most famous is Oktoberfest).

The time frame you mentioned would meet your weather requirements pretty well. Temperatures will likely be mild to slightly warm. Don't worry about "crowds". Certain attractions might individually see large numbers of visitors (ie, Neuschwanstein, Miniatur Wunderland), but an entire country is never going to feel like, for example, a busy day at Disney World.

Either May/June or September/October... In both cases nice weather, outside seating but no high season, no real crowds.

i went around octoberfest time and it was pretty nice. just a jacket over t-shirt, very comfortable. i went to munich on a few weekdays to avoid crowd but to sample the festival. if you like driving you can rent a car, which allowed me to visit many small castles and villages and stayed at farm houses at lower cost.

The best time to go to Europe would be before June and September/October. We usually go the last of March thru mid-April. May is also great. Airfares and room prices are the highest during Summer. Summer temperatures are high, and air conditioning may not be up to your Florida standards. And massive numbers of Summer tourists can be avoided.
In Germany, there are no dead months to visit. Some months are just more manageable and pleasurable than others.