According to the Western Hemisphere Travel Initiative, a "Closed Loop Cruise" that is, one that starts and ends at the same US Port, can be taken with just a Driver's License and Birth Certificate. That said, if you plan on cruising a lot (and you will) go ahead and get the Passport. You probably don't really have enough time to get one before this cruise, but tell Santa you want one for Christmas.
As to the dress. Should be no problem. The formal night really is not like a Prom. Church clothes often work.
Enjoy your cruise.