..from my understanding after you take the DAT your scores are distributed to schools chosen by you without your participation. So yeah, if I understand your question right, all you do after taking the test is.. wait
I called AADSAS a couple weeks ago and asked that very question. I was told to update my application with my unofficial scores if I so desire and they will be sent out in the next mailing cycle as unofficial. Since official scores only come from the ADA, the delay you may have heard about has nothing to do with updating your application.