If you're asking this question now there is very little reason why you shouldn't be able to do it much earlier. Don't wait on your DAT to submit. And don't wait on LORs to submit. Really, it would be foolish to wait that long intentionally. All you need done are the personal statement, school choices, extra curricular summaries and transcripts to submit. The schools won't get your application from AADSAS until a month after you hit submit. Add the other stuff later.