I agree, drop family and luggage first. Just be sure that you keep YOUR travel documents and ID with you so that you can get back into the port area.
The advantages of dropping family first are that you only have to handle the luggage once, Mom can get in line and be moving up in the check in line while you are returning car (you will join her in the line when you get back), and you will be able to catch the first shuttle from the car place to the port. It's easy to fit one person, no luggage into a shuttle. Getting a family with luggage onto one may mean waiting....