I was in the Brownsville area, actually at McAllen, at this time 3 years ago - birding was terrific. I bought Ken Kaufman's Birds of N.America on my first morning at the Benson Rio Grand Valley Park as so many birds are specific to S. Texas only. The greenjays were gorgeous and then there were chacalacas, clay-coloured robins, doves - inca, white wing and white tipped (rare, S.Texas only,) and many others.
Santa Ana Wildlife Refuge; kiskadees, Couch's kingbirds, golden fronted woodpeckers. Stilts, dabbling ducks.
Quinta Mazatlan (closed Mondays). Lucked out on a long billed thrasher here. Many warblers and woodpeckers.
Weslaco Estero Llano Grand State Park: good for wading birds - special for me here, white faced ibis (similar to glossy). Also vermilion flycatcher, often at Grebe Pond. At the alligator pond - both Green and Ringed kingfishers.
When you're in the area you will find pamphlets on many other areas.
As someone has pointed out, the Texas Coast Birding Trail is v. long. I've had good success in the Rockport area and of course the whooping cranes are in the Aransas area.
Enjoy.