No July 8th is not late. I took mine at the end of July and received two hands filled of interviews. What's late? Approaching the end of August and into September and thereafter. Remember that it takes 4-6 weeks for AADSAS to process your application and about 2 weeks for your official DAT scores to be revealed.
If you take your DAT July 8th then try to submit your AADSAS application in June indicating that you will take your DAT July 2013. Then by the time your DAT rolls around, your AADSAS app will have been processed (as long as you submitted it 4-6 weeks before your DAT) and then all you have to wait for is your official DAT scores (2 week waiting period). In the meantime, you can fax/scan your unofficial scores to the schools you are most interested in. Then go enjoy your summer. 😀