Applying for a job as a couple is not unheard of, but not the ideal position, either. However, my guess is that if you both express an interest in working hard NCLA would be happy to talk to you and if they like you might hire you. If they do, you would definitely be given a cabin for just the two of you.
You sound pretty young, but I wouldn't settle for the lowest paying position if I were you. I would suggest the two of you get some experience working as waiters for 6 months, and then apply for that job.
Or room steward is always an alternative.
I will go out on a limb, assuming you are not an entertainer or certified sailor, and say that bartending would probably be the ideal job in terms of pay and hours. But they want experienced bartenders, so you would need to get some training and experience first.
Here is my worry, that if you went to NCLA and said you would take a dishwashing job just because you want to travel that they would see you becoming one of the crewmembers they have the most problems with - the ones who come on board thinking it is going to be a "fun job" with "travel" and end up finding out doing the same itinerary for 4 months is not exactly "travel," it is more like "relocating to Hawaii." Those people become disillusioned with the job and often become the complainers.
What they want are people with realistic expectations of the job they are walking into, and are doing it to work and save money, not just for a chance to get on a cruise ship.