Yes you can still apply now. You can submit your primary so that it will be verified by the time that you get your score. Once you do get the score, AACOMAS will send out an update to all of the schools that you have applied to.
The benefit of waiting however is that you will now what your score is and how broadly you need to apply. If you want to submit before you get your score I would recommend applying broadly. But since you just took the MCAT today you will get your score in a month so submitting in June isn't late at all.
I read on here somewhere ... apply now but to only one school. This will get you verified ... which can take up to six weeks. Once scores are in then apply to the rest.