There is so much to see and do in San Antonio--the Missions (there are more than just the Alamo), El Mercado (shopping on a Mexican street without going to Mexico), the Pearl District with shops and restaurants, and lovely parks if you want to get outside and relax.

https://www.visitsanantonio.com will give you an idea of the many attractions.

I'd skip Austin.

Also like The Japanese Tea Garden and The Saga. Botanical Gardens are good if you're up for a lot of walking and the weather is nice. Might be better in March/April/May when more things are in bloom. The narrated boat tours on the river are fun too. Have enjoyed a Segway tour as well.