Cheryl,
I have always sailed out of Galveston on all the cruises I have been on. I have stayed in Houston and near Galveston. I have not stayed in Galveston as I find the hotels are usually unavailable or expensive. I did stay once in Texas City across the bay one time. I stayed at a Marriott Fairfield (I think) - Anyway, it was Marriott, because I used points to pay for the room. It was a nice hotel and just 16 miles away from Galveston. It was a nice place and I felt very safe there. It was not in the heart of the refineries, but near a Mall that did not seem very busy. There were places to eat and I did not get lost.
Anyway, this may not be the answer you were looking for, but it can be an option for you.
Plan on about 50-60 dollars to park each car for the week at the parking lot by the pier.
Let me know if I can help more. . . I am sure there a lot of people who will have good feedback for you on this one.