Dolphins live along the entire Texas Coast, with prime viewing spots from the ferries that run between Port Bolivar and Galveston and between Aransas Pass and Port Aransas. Both the dolphins and the ferries are delights. Hundreds of ocean-going ships a year pass next to the Galveston east jetty at the east end of Seawall Boulevard, where a clear view is possible. The north jetties at Port Aransas offer a view of ships entering and leaving Corpus Christi Bay. A few Galveston and Corpus Christi wharves, where ships load and unload, are also fairly accessible for viewing.
Shrimp boats are so common that they're hard to miss.
Beaches in urban areas typically don't allow dogs or they require that dogs be leashed, but you probably won't have to go too far to find an open beach with no prohibition. If you plan to stay at a motel, check ahead to make sure pets are allowed. Few motel operators welcome wet, sandy dogs. Check with the city parks department at your intended destination. Corpus Christi and Galveston have especially nice museums and other diversions for children if the beach gets boring or stormy.