Here is a place in Corpus Christi that has bareboat charters. I have no experience with them.
Yachting Center of Corpus Christi
In this area, you have anchorages around Corpus Christi Bay and the surrounding bays.
You can visit the Matagorda Island State Park. It is a closed state park but there are docks or you can anchor (recommended as there are lots of raccoons that will find their way onto your boat). This give you access to a remote stretch of Matagorda Island with beaches and an old runway. It is likely you will not see anyone else.
You can marina hop and go to Rockport, Port Aransas, Ingleside, etc. If you time it right, you can go to little festivals at some of the towns.
Anchoring out, you have good crabbing off your boat and have good fishing with a dinghy or kayak. There are a number of kayak trails such as Lighthouse Lakes and Mustang Island (TPWD: Coastal Trails | Texas Paddling Trails
). You can sail to an anchorage and then kayak parts of the trails at your leisure for recreation, birdwatching, or fishing.