Short answer - yes, you can order from room service for the dinners that are being served in the dining room.
You and your family might just enjoy having these dinners in the dining room - either the Traditional dining if you've booked it ... or in Personal choice.
As to being overly crowded - I doubt it. Sailed on the Caribbean Princess last November and didn't feel so. The ship is broken up into smaller spaces so you don't feel crowded.