No, you can check the test dates without applying to take the test. Google "locate a test center Prometric", locate ADA as the testing sponsor, and click "I want to locate a test center" (green button in the middle).
Enter your state, choose DAT as the exam, and then enter your address. You can choose "availability" after you locate a specific testing site and a then a calendar will appear with all of the available test dates and times for that center.
Most people recommend booking your test around 2 months in advance so that you can get the location and time that you want. Also note that your application to take the test takes around a day or two to be processed.