Hi Deb,
This is a great question as I, for one, am always trying to research when is the optimum time to visit the Walt Disney World Hotel.
There are multiple "seasons" each year. Value Season is the least expensive and least crowded time, followed by Fall Season, Regular Season, Summer Season, Peak Season and Holiday Season.
Late April/early May is considered to be Regular Season so you can expect to find moderate crowd levels when visiting at this time of year.
The unofficial website www.mousesavers.com has a list of the Room Rates and Season Dates for every Disney Resort hotel throughout 2016 & 2017.
In terms of which Disney Resort hotel would be the best for children aged 11 & 13-years old, that really depends on their individual interests and your budget. I would recommend taking a look at the different
Places to Stay on the Walt Disney World website to narrow your choices down. I can assure you that no matter which Disney Resort hotel you choose, your children will have a blast. My family have stayed at many of the Value, Moderate, Deluxe Resort hotels as well as the Deluxe Villas over the years and my son has thoroughly enjoyed his stay at every single one of them.
Wishing you a magical trip filled with memories which will last a lifetime!
Emma