This is not that complicated.
1. Schedule the DAT and start studying.
2. ask for LORs now so that you'd get them when the cycle begins.
3. Continue shadowing, but don't kill yourself over this.
4. Start writing your PS.
5. When the next cycle begins, fill out your AADSAS (transcript, LORs, and etc.) and submit as soon as they receive your DAT score.
6. Sit tight and PROFIT! I mean kill those interviews.
If you're taking the DAT in July/Aug, you still have a good 3-4 months to get more shadowing hours. Just do well on your DAT and you're golden.