A good reference is http://www.petswelcome.com...">I've http://www.petswelcome.com, I've used it a number of times and you'd be surprised at which hotels accept pets. You can plug in your location and go from there.
Best of luck,
I don't know what part of the Texas Gulf Coast you're looking at, but the Upper Texas Gulf Coast (east of Galveston) is the "Bolivar Peninsula". Here, there is no barrier island, unlike most of the Gulf Coast. My family used to vacation in rental beach houses on the Bolivar Peninsula every summer, many years ago.
Otherwise, there are few places without a barrier island, except for near Lake Jackson, but I've never been there...Good luck!
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