We've sailed out of Galveston twice and didn't have any "hassles"
Where to stay really depends on how much you want to spend. The prices in Galveston can fluctuate greatly, depending on what's going on at the time
We've stayed at Commodore on the Beach (check out their website). It's an older motel, but clean, the people are friendly, and it's right across from the beach.
You can leave your car in their lot for free during your cruise. It's not secured, but they do monitor the area from a closed circuit TV at the desk. They don't have a shuttle to the pier, but will call you a cab (I think it was $8.00) when you're ready to go.
Many people recommend the Harbor House, so you might want to check their site, too. I believe that Harbor House is in walking distance of the pier if you can handle your luggage yourself
, but I don't know about parking there.
If you have time, be sure to go to the Strand area. Lots of great shops! Or just "cruise" up and down Seawall Blvd. and people watch! Galveston is small enough that it's almost impossible to get lost.