I wouldn't ride Alafia in the wet at all. If I get caught in the rain there I'd just go back to the lot and call it a day. The trails don't hold up well to wet riding and the wet clay is like ice. If the area has had torrential rains I would avoid it for at least a few days to let it dry out.
Boyette for the most part does okay it seems but I would not ride the more difficult trails unless you give it time to dry.
WT is great. During the summer you will likely see some trails closed as they tend to flood easily. I've been caught in some monsoon conditions at WT where it felt more like river biking on the main trail, but they drain quickly. There are, however, some spots on the main trail in the center of the paved loop that can get extremely flooded if we get a ton of rain. I would avoid those when conditions apply.
Croom, on the other hand, can pretty much be ridden year round. Avoid the optional hard trails and enjoy the yellow/blue trails if we have had a ton of rain.