Depends on you and your family, but I do put a value on early entry. We usually take advantage of EE, especially at the MK. We sometimes switch parks midday, but don't always. I would switch parks if I was taking a mid-day break, anyway or if the first park got real busy.
This last trip (just my DH and myself) we only did one EE and that was because we wanted to go to MGM on our last day and it happened to have EE that day. We did evening activities, unlike when we have our kids with us, so didn't want to wake up that extra hour early.
Are you and your kids normally early risers? June will be hot so it might be nice to go to the EE park, stay several hours, take a break, and then go back to a different park. That way you'll be in the room during the hottest part of the day. I'm assuming with kids who are 4 and 7 you won't be out until late in the evening.
Personally, with only 4 in the family I would stay onsite. You can stay in any resort and can try to secure a discount rate.
T&B