I booked Pop Century for May 31 through June 3 just a few days ago. I didn't have a problem, though I didn't look at any other resorts. Disney does have a Florida resident special going on, which may be part of the reason for the challenge.
I've taken a trip to Disney the first week of June the last few years. My daughter's summer break starts at the end of May so we head down the day after Memorial Day. It's a bit crowded, though not bad, particularly in the evening. We barely wait in line at the water parks. Give it another couple of weeks and it will be a mad house I'm sure.