We go to Disney every Easter, pretty much. We also go to Disney the last week in August (we have annual passes....). From our experience, when Easter falls at the end of March/very early April, the crowds are heavier than when Easter falls in late April (like this year). If your family can deal with crowds, you would be ok. We love it, no matter how crowded, but we go to Disney a lot and don't feel the need to do everything each trip. With the utilization of Fast Pass Plus, you'd get to have some fast passes for your trip, so at least you'd be guaranteed to get on your must-do rides. There are kinks in the system right now since it's all new, but by the time you go hopefully FP+ will be of great benefit.
The end of August.....we pick that time because of the huge difference in cost and it's refreshing to actually have a lower-priced trip. However, we don't enjoy the weather, not because of the heat but more because it rains pretty much every day, even if just a short downfall, but some days it lasts longer than that. We find it to be a nuisance, no matter what we're doing whether it's at the pool or walking thru a park.
I'm the outcast on this, I know, but I'd probably go with Easter time frame. But only if your family can handle the crowds. We much prefer the weather at that time of year, and we don't mind the crowds. Our all-time favorite time to go to Disney used to be in May because the weather was perfect that month, but with the kids in school now, we need to pick school breaks which is what I assume you guys are doing.... If the 4th of July week is an option, that was actually a great time to go

. We didn't get a lot of rain last year and there were room discounts for that week too. Good luck!