We visited Aransas National Wildlife refuge on one December afternoon. There are plenty of small trails and a variety of wildlife. We saw alligators just off a trail (see picture) - they all seemed to be sleeping in the afternoon. We also startled a javelina.. it came running out of the thicket, saw us, and ran straight back! There are a few observatories for bird watching. Make sure you take your binoculars, or you can also rent them out at the office.
We just had about 3 hours, and felt like we needed at least a day to spend there.