Welcome to the area, Sorry to say that most of the land around Del Rio is private and there is very little off roading there. However, there is a lot of things to do in the area with the Lake (a good SCUBA and fishing lake) and a rich history.
Your best bet is the Big Bend area. But I will caution you that it is HOT in the the summer. Was down there mid March and temps in low areas was already 3 digits
I hit that area in the winter months when it is more like summer there and everywhere else cold. Winters are very cold nights and 60-70 degree days. And use the summers for Colorado and New Mexico.
Attached is a video of photos from 2011 with my CJ. (now have a Rubicon, but have not organized photos with trips in the new(er) jeep).