P&O Ventura and Oceana are the best ships for your cruise.
I would also have to recommend Royal Caribbean for teenagers, especially Voyager and Freedom Class ships. Independance of the Seas sails out of Southampton starting in late April next year for med and canaries cruises as well as some to northern Europe and Scandinavia. The facilities on board these ships are excellent with ice rink, surfing, rock climbing, basketball, volleyball plus arcade games and teenagers club. Movies run constantly on the cabin TV system and there is internet facilities at a price. During the day they will all congregate aft on the upperdeck and leave you to relax in relative peace. Avoid P & O, Fred Olsen, Cunard. They are not really suited to teenagers. Avoid NCL also - they are not suited to reasonably mature adults.
I would go along with Rum Runner and Wilba but would include RCI’s Radiance Class because you said you would like a bit of class and that range of ships offers most of what is offered by Voyager and Freedom without the numbers of people and with a style that is so beautifully different. The children, well you'll be surprised how little you see of them, RCI have the best staff for keeping them active and amused and that is why they are so rarely a problem as they are on so many other cruise lines we could mention. Shame you don't want to fly though because it opens up a brand new world, mind you, you could always drive down to Barcelona but with children that is so much hassle. Regards,,,, Neil.
Would chose RCI from Southampton or Harwich but may be consider P&O with Oceana it seems to get good reviews and is a family ship the tips are not added to drinks(as yet) and you are only expected to tip the waiter and cabin attendant drink prices cheaper than RCI but service level not as good ,then why not look at Celebrity bit dearer than RCI but upmarket ?