This past Saturday night there were no seats open within 100 miles of Chicago, but two seats three hours away in Indiana. Within half an hour, one of those seats was gone (they were at two different locations, but close to each other). So, I took the other seat. From Saturday night until Monday morning AAMC kept saying that my change was being processed and wouldn't even let me search to see if a Chicago seat had opened up despite the fact that I'd already received the email confirming my change. I'm not sure if this is usual (i.e., if you make a change during the weekend your registration is locked out from additional changes until Monday), or if there were problems this weekend. Anyway, I looked early this morning around 8:30am CST and there was an 8am and a 1:30pm seat open in Chicago for June 15th. Perhaps seats that open up over the weekend don't get released until Monday morning? I don't know.
So, my suggestion is to just keep checking. I'm sure something will open up eventually. Also, once I registered for June 15 it said that I had until June 1st to make any changes. So, I'd guess that people who decide to put it off to July will be looking to do that before June 1st.... Good luck.