The beach areas are not particularly birdy. The Twin Bridges Trail has had some water birds, ducks and herons. There have been Great Horned Owls calling near the southern turn into the Horseshoe Camping Loop at dusk.

Check in at the main park entrance first to obtain your entrance permit. There is a per-person entrance fee to use the state park. A state park pass might be a wise investment if multiple people are birding or if you think you will be accessing a Texas State Park multiple times over the next 12 months. Check here to see if a state parks pass is right for you: https://tpwd.texas.gov/state-parks/park-information/passes/park-passes

Twin Bridges Trail is the best trail for birds. 

Another trail in the Park, the Towash Forest Trail has a side trail that branches off to the right, a short distance (100 yards?) from the trailhead. This branch trail goes down to the lake, but is a dead end. The Towash Trail has a loop at the end that also goes by the lake.