HHI is pretty much booked to capacity entire summer; we are on waitlist for 3 nights in August
FOREVER 
There's some cash inventory but I don't really want to do cash as it is around $352 per night for 2br. idk about June because we usually go in July & August but I think it may be slightly less crowded except for maybe late in August.
We have never had an issue with storms, but maybe we have just been really lucky

It is very HUMID though, temps high 80's into 90's. Even with a storm coming up the Atlantic coast HHI usually doesn't take a direct hit because of how it's situated. Here's a link to temperature & weather info
http://www.hiltonheadisland.org/about-the-island/weather/
DH & I & our

(ds 22, dd 19 & dd 13) LOVE HHI; the kids actually wanted to skip WDW this summer and just go to HHI but then some of their friends & my DBro & his family will be at WDW so we kept those plans but we will still stop at HHI on our drive to/from. We go to VB too but that's just on our " like " list, but HHI

IT !!! Restaurants AWESOME & plenty of activities both at Disney resort & nearby, our girls love all the shops on island too. We're going to try to get a day trip in to Savannah this summer also.
It's ALOT different atmosphere than VB but it still has a laid back, relaxing vibe to it but not that isolated - lights out at 9pm feel you kinda get at VB (LOL !)

Laura