For non-holiday or spring-break sailings, Princess does limit the number of children on the ship under a certain age, but I think the cut-off may be 12, so you should be safe. Plus, you would be going at a time when most families aren't, as their are waiting a few weeks for school to let out, or went earlier during spring break.
On the Golden for the 3/22, my friend said that there were way too many young children running around without supervision. They were knocking on cabin doors and running away; pressing the buttons in the elevators so they stopped on every floor, etc. We've been on two sailings on the Grand and never experienced anything like this, but then we've always cruised in May, not March/April...prime spring break time.