Stephanie,
It depends.
If you really don't feel ready, then yes I would recommend that you wait. There is no point in attempting the exam if you personally think that you are going to fail as you go into it. That would be a self fulfilling prophecy.
Also remember that the last day of the old exam is on the 30th - everyone is now trying to get their exam done. So if you have not yet scheduled your exam and you already have a date then it is very likely that all the open spots are already taken. And because of that you may be forced to wait until a seat is available after June 30th to take the new exam.
Furthermore, remember that the PMP exam doesn't really change significantly after June 30th. There are areas that will change (i.e. on questions like What are the inputs to the Create WBS process") but the general concepts of project management have still remained the same, the earned value formulas are still the same and a project manager is still supposed to be proactive. So you will very likely encounter many of the exact same questions on July 1st as you would on June 30th. It's an evolutionary and not a revolutionary change.