This was our third cruise on the Rhapsody and I'm sorry to see that the ship will be leaving Galveston later this year (September I believe).
Leaving Galveston has always been very simple for us. We live in Houston so knowing our way around the port is not a challenge. However, there are typically 2 ships leaving on Sundays. I would suggest that you arrive early, drop your luggage at the port and then park your car and take the free shuttle back.
Always have your documentation ready before going to the ship. It just makes it all go so quickly. Since we have always been in balcony cabins, we have never waited more than 15-20 minutes to get on the ship.