During the eclipse do I have to keep the app on the either of the two main timer screens: Total or Partial? 
Yes. I recommend staying the the either of the two main timer screens during the eclipse.  These screens provide you with the most inforamtion about what is happening with the eclipse.  In addition, Apple and Android handle the audio play back differently if the the app is not a timer screen. Especially Android.  Being on another screen has the risk of making you miss audio annoucements.