In our case, kids 3 and 7 at the time, we got most of our stuff ready the night before and after checking out in the morning we made it a park day. As someone said above, make it a park that is closer to your new resort. In your case, HS or Epcot. It was not an issue for us or the kids. If the room isn't ready at the earlier time, you have the park to entertain OR the new resort yourself is yours to enjoy. Honestly, you can use the pool at the old resort the rest of checkout day.
It just depends on your family. If the 4-year-old naps religiously every day no matter the setting, then you might have issues. Other than that, you should have no issues! Enjoy!