Don't rush and make silly mistakes. Once you hit submit there is no going back. For instance errors in the personal statement letter, etc. Also double triple check that each and ever course that you submit in the course work section of your application matches EXACTLY what is in your school's transcript. AADSAS verifies and calculates your GPA, if the course work you submitted does not match your transcript, they will send it back to you and tell you to fix it. This will prolong the application time.
Whether you hit submit on June 3rd or June 10th, is NOT going to make much of a difference if any. Applications are mailed in batches. It may be likely that people who submitted on June 3rd and June 10th are included in the same batch. Even if they are not, schools do not start reviewing applications until August anyways.