It just depends on what you have going on when you are trying to study. I would highly recommend studying for the DAT alone versus trying to balance it with a semester of classes. In my opinion the best time to take the DAT is during the first 6-8 weeks of the summer after spring semester. You are fresh out of classes and theoretically that material should be relatively accesible in your brain which could reduce the amount of studying you need. For example, I studying for the DAT right after finishing a year of organic and bio and that material was really fresh and minimized the material I needed to reivew. Thus, I could focus on math, GC, PAT etc. instead and bring up those scores.
Studying for the DAT while taking classes is such a waste of time IMO. You cannot truly focus on either and both your grades and potential DAT performance will suffer. Plus, there will be burnout etc., for studying over four months. Just crush 8 hours a day for 6 weeks over the summer and knock it out....
Also, please don't believe the hype over applying the second the application cycle opens. Anything submitted to AADSAS in June/July is still very early and not detrimental to your chances of getting pre-december interviews or acceptances. While you are studying for the DAT take an hour break and work on your application during the summer. Then, when you take the DAT and it is acceptable send your application. By the time AADSAS receives your scores you will be ready for mailing.
I took the DAT July 1st and submitted my app mid-July and was mailed out to schools end of July. However, my packet wasn't done as I was waiting on a committee letter until end of August and THEN my application was complete on AADSAS. Even so I got 7 interviews....this goes to show that an application complete end of August is still in play...