You can call 407-WDW-DINE while you're there, or stop by your hotel concierge desk and ask them to see what's available for that day.
I still wouldn't go without ADRs. Many many schools are out already, but yes, not all. In the south, probably most of the public schools are finishing up this week.
As for Gay Days, that has a minimal impact on overall crowds (Gay Days has a designated park each day) and I think even less on dining. That shouldn't be an issue, but general summer vacation crowds will be.