You will need to take your luggage with you as ships rarely have space for all their necessities as it is and space costs money with any business. Also, they would not want to be held responsible.
Phone Cunard direct. I know that it is possible to leave luggage on board P&O World Cruises for a charge. However they start and finish in Southampton and this may make a difference. Sometimes Cunard World Cruises start and end in the US.
Due to security cunard no longer allow passengers to leave their luggage onboard. the do offer 'whitestar luggage service' which will courier your bags to your front door, telephone number 0800 051641.
Unfortunately it is never possible to put unaccompanied luggage on Cunard ships. unlike P&O we don't do that service. We do have a service called White Star Luggage Service which is operated by DHL. The number is available from 2pm-9pm and is 0800 051 6414.