Based on your requirements, the best times to cruise are the 2nd - 3rd week in September. This is pretty much true just about anywhere you wan to go. But kids are back to school, parents are back to work, and no one wants to cruise to the Caribbean because it's the height of hurricane season. But that's when we like to travel as we get the best rates and hardly any kids.
Also, the 1st - 2nd week in December is good. People like to cruise during Thanksgiving and Christmas, but they're not planning to cruise between the holidays because they're getting ready for Christmas and buying gifts. So they're not thinking about spending money on vacations.
As for when they are older teens and not younger teens, just doesn't happen because teens are teens when it comes to when they can cruise with their families.
Now, with that said, the later you go in August, the less there will be because is some States, school starts earlier than others.
Unfortunately, families with teens are very limited as to when they can cruise due to school. Cruise lines know when school is out because that's when the biggest demand is, so they're going to have higher prices at those times.