Hmm okay so if they're due in May, that means you can submit your final transcript grades to AMCAS on time right? Because I heard that for people on a quarter system, it's a mistake to wait for their final grades to come out in June (before submitting their app) as it causes a huge delay...
It depends on how late in June. If it is late, I would say it is absolutely not worth it and you should instead send your schools updated transcripts after you apply. Early June is more of a variable area. I'd say it depends on GPA in those cases.
The spring semesters at my school end the last week of April with the first week of May being finals. Grades were generally due within the week after finals (earlier for seniors).