I believe that you should at least get your DAT scores in before you submit. Although, your score might not be official until the school receives the actual scores from the ADA, they will still be published on your application, which is VERY important.
Anyways, I'll explain what I did for the entire process.
1) Long before May 15, I archived my LORs in an "official" professional school LOR archiving service that was associated with my school. This way, I could request that they be sent out all at once.
2) I completed my DAT, but didn't send out scores before May 15. There's really no point in sending them out before May 15 because they might get lost.
3) On May 15 (well, actually May 20, but close enough), I submitted my application and immediately printed out my transcript matching form. I gave this to the registrar at my undergrad institution and had them mail out my transcripts with the matching form ASAP.
3) Also on May 15, I entered the info of the people who wrote recs for me on AADSAS and printed out the matching forms. I then gave these all to my undergrad institution's LOR archiving service and had them mail out my LORs to AADSAS right away.
4) Also on May 15, I put in an online request to send out all of my DAT scores to the schools I applied to.
There are you are! Voila! This is the absolute fastest way to do everything. I might mention that it's critical to get your transcripts in as fast as possible because the grade verification service really gets bogged down as the summer progresses... and they won't mail your app to the dental schools until your AADsAS GPA has been calculated. If you submit on May 15, the turnaround can be as little as 2 weeks. Towards the middle of June, it can be as long as 4-6. This is a very serious difference.