We put our sons in their own inside cabin across the hall from us last year, and will do so again on our next cruise in 2010 (they will be 18 and 14 just like yours). Your situation may be a little more complicated with a boy and a girl, especially since they don't want to share. In 2005, we did an adjoining room with a shared common area on board a Princess ship.
The cruiseline won't hassle your kids, as long as they aren't acting unsafely or harrassing other passengers. They're free to roam the ship, and no one will bother them. There is usually an arcade, but that costs extra so you may want to discuss limits.
One thing we're doing next year to keep them occupied, is SCUBA lessons. They'll go through the classroom on our first sea day, and do open water certification in Curacao. We're also taking a friend of my older son's, so all three of the boys will do this together.