Hi, daytime in the Caribbean? You won't need long pants for that......lightweight capris and shorts are what I usually wear during the day......if you are on the ship? Bathing suit and cover up always works. If you are more comfortable in a sundress, that works too.
I have never sailed with Princess but I have been cruising for over 20 years on many different lines........a Caribbean cruise, (no matter the line) is really casual, at least during the day. If you feel like dressing up for dinner? You may certainly do so. I wear slacks and nice tops every night.....I am not a "dress" gal so nicer slacks work for me.
Hope you both have a great cruise๐