You can do it if you get very pro-active. Don't send anything regular mail - make it priority or overnight. Fax in unofficial copies of DAT scores ahead of time if they aren't already on your application. Figure out which schools have secondaries that you can start working on right away and get moving. Some schools require an additional application fee that they want ASAP - figure out which schools these are and send in your checks. Some others will want an official copy of your transcripts - these can take some time, depending on how efficiently your transcript office works, so get that started. Call schools to verify that they received everything, and call again to make sure your file is complete.
And whatever you do, don't rely on AADSAS to tell you your application status!