As the above poster said, hard to say exactly without a DAT score. If you do well, you can afford to be a little more picky. If you don't do so well, then applying to more schools may benefit you.
How about right now, you only apply to your top 10 schools. Then, once you know your scores, add more schools if you need to. There's no need to apply to a TON right now, especially since they likely won't even look at your application until your DAT scores are in. Even if you add schools at a later date, you will still pay the reduced rate (which I think might be $93) for the additional schools.
Try to take the DAT before September. Any way you could make it happen in early or mid August? For one, it'll put you at an earlier time... and it would also mean that you wouldn't have to be studying for the DAT once school starts back. If anything, take it the day before you fall classes begin. I wouldn't want to worry about both at the same time.