If it was me I would head out west toward the hill country.
With a 1 yr old and the Texas Summer heat I would not want to be doing a lof of hiking and some times state parks don't have the best swimming options but thats just me.
I recommend the 7bluff cabins along the Frio River. Nice calm pools for the baby to be able to play in, shade trees nice cabins just a nice relaxing time overall. We have stayed here before a really nice place.
And if you do want to do some hiking you are 5 minutes from garner state park. 7Bluffs
This post was edited on 3/7 at 8:38 am