I use to live in that area. Now I don't know about hotels that will let you leave your car. I do know about the good hotels on the Island. What type of price range are you looking for?
I personal like the Galvez http://www.wyndham.com/hotels/GLSHG/main.wnt
Sadly in april you are in the spring break period so the room rates go up on the island quite a bit every were. The rooms at the Galvez will be over 200 bucks in April were as right now they are about $125 this time of year.
To be honest I would not stay on the Island that time of year. I would stay. There are some hotels in Texas city and the Clear lake area. They will be cheaper than the Island that time of year. If you stay off the Island you would want to leave early to get to the port you never know when the causeway traffic is going to be backed up.
We've cruised out of Galveston three times and have found the Days Inn to be a great one for our needs.
They let us park our vehicles there for free, and we've never had any problems in leaving them there.
I don't know if they have a shuttle, we've never asked. We take everyone and all the luggage over to the terminal, and then just one driver for each take the vehicles back to the hotel and a taxi back to the terminal. It sounds like a lot, but the whole process only takes about a half an hour, and the taxi cost is about $12.
We stayed at LaQuinta and we parked our car for $10.00 for the week. They have a shuttle to the port. We were told the $10.00 fee was a one time only fee, if you did future cruises the parking and shuttle is free. The shuttle was free this time also. The room was reasonable and clean. They also gave excellent recommendations on restaurants in the area.
We cruised out of Galveston three weeks ago and stayed at the Ramada Limited, NOT what I would call a good hotel in the Ramada chain....barely a two star out of four stars. They supposedly do have cruise parking, but we took a cab and it costed a whole $5 and we gave a decent tip because we had so many bags (7 suitcases and three carry ons for three people!!! My bags came back stuffed full...).
Hotel Galvez looks amazing. We walked down Seawall on the morning of the cruise and walked past it...wow, it looks nice, alas, it was $250/night plus tax if we were going to book it for the precruise stay...
Harbour House and Tremont were were I wanted to stay, but we couldn't get availibility for one night stay, two yes, but not one.
2007 Victory & Triumph
2008 Conquest & Liberty
2009 Valor, Freedom, and Splendor
2010 Glory and Liberty