I wouldn't worry about taking the DAT at the end of June. If you are able to stick with that test date, then your scores should be processed and your application complete and ready for review by the third week of July, which is still relatively early. Just focus on scoring as high as possible and making sure the rest of your application is strong. Don't forget to remind those who are writing letters of rec for you to submit them once the application opens, so you have those ready. Start the transcript entry and verification process early, as that can take 2-3 weeks to complete. Ensure that you have a strong personal statement, and have a few other people read it. By staying on top of the rest of the application early on, you will be able to focus on studying for the DAT as your test date approaches as opposed to scrambling to finish your application. Good luck on the DAT and on the application cycle!