Well it takes 2-4 weeks from the ADA, so you should be fine there, but then consider the ADA sends them out in batches, not as they finish verifying them.
Not sure if you read the message in your AADSAS application:
Please be aware that AADSAS is in the process of receiving and processing DAT scores from the ADA. If AADSAS has received your DAT score from the ADA, you may see it posted to your application as early as June 23, 2011. AADSAS will continue to receive scores after this date and post them to applications as they are received. Please be patient while these scores are posted.
As for submitting, I'd just do it as soon as you completed everything. Your application will be updated with your DAT scores automatically when they do get them.