Look at Park and Go. Be sure to look at the cruise lot and not the airport. They drive you to the port and pick up. Lot is next to port. You take your luggage with you on their bus so no need to drop off first at port. I think I would be at the lot at least an hour before you want to board the ship. They will go to a few ships on each trip. I think the rate is cheaper if you sign up on line and you will also have a reserved spot. They park your car so you have to leave the keys.
Park 'N Go - Port Everglades Only Parking (FLZ) Port Everglades Reservations & Reviews