As far as I know the only one is the Disney cruise line. My son and his friends did it during spring break 2004 and had a great time. They did the 4 night cruise out of Port Canaveral. Since the Disney crusie line is more expensive they got an inside cabin and said it was fine. You're not in the cabin that much anyway. Check it out .
On Princess, anyone under 18 must travel with someone 21 and over. If you are already 18, you shouldn't have a problem on Princess.
CarnivalSensation February 2015
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Carnival Fascination April 2014
Carnival Elation March 2011
Carnival Imagination Sept 2007
Carnival Sensation Dec 2006
RCI Sovereign of the Seas Sept 2006
Carnival Miracle Sept 2005
Carnival Glory Sept 2004
Carnival Fantasy Jan 2004