DD & i went during the first week of june last year. definately
not a slow time.
btw, we avoided whichever was the "park of the day" posted on the website for gay week, figuring crowds would be lower at the other parks. would've been a great plan except that it is simply a moderately busy time of year

(for us, it being "gay days" was a non-issue except for the crowds).
mostly the daytime weather was beautiful (warm, but not hot), with some rainy days/nights, as well as one night where it literally poured for hours (not being able to see more than a foot or two "pouring"), & became
very chilly.
we've gone repeatedly in aug thru oct, & i think it rained more in that trip than any of the other ones in "hurricane season"
we just took advantage of EMHs (parks were gloriously empty at night

), and actually used ponchos

lol.
we (DH & i) are going the first week of may, & are hoping for lower crowds (like the "good old days"

)