As a 3X applicant that has done things wrong and right, this is what i would suggest-
first of all.... ONLY TAKE THE MCAT WHEN YOU ARE READY!
If this is your second+ time taking the MCAT, taking a June MCAT is fine because your AMCAS can still be verified while you wait for your new score (probably the most important step based on the 2013 application cycle disaster of 8 weeks to verify). If you do this, this is what you need to do.:Turn in your primary the first day it opens so you get verified in a timely manner. Take your MCAT (Ideally I wouldn't take it any later than the end of July only if you have if you have great numbers and have a previous MCAT score). Write all of your secondaries, because schools will immediately get your MCAT score the day it is released (I got my MCAT score when I was filling out the BU secondary lol) and schools will hold your secondary until they get your new score.
If this is your first MCAT, taking a june date is a little risky because that means that your primary will not be considered complete and ready for verification until July. Generally there is a trend that the later you submit your primary, the longer it takes to get verified. This year and 2012 were normal, but last year the verification process was about 2 months if you submitted after June 10th. If you have any weakness in your application, it is essential that you turn in your apps as early as possible because it is rolling, and pre-meds are in general very type A and are pretty neurotic, so you can bet that most of the other 45K applications will be applying on time.
I would plan on studying for the May date, since you said that is the one you would like to take. I know that a lot of states are sold out right now, but what happens is that as you get closer to the actual test date people start to freak out, and will push their test back. Generally this happens about a month before the test. Last year I was signed up for for May 17th, but ended up freaking out around the end of April after 1 particularly horrible practice test, and canceled my test. I was able to sign up for June 13th at my first choice location in California (California sells out within the first day registration opens everyday).
I hope this helps! best of luck!