Roadie, MTB, either/both?
Family vacation in Texas says "San Antonio" to me, but since that's where you live, it isn't an option. Eight hours in a car doesn't get you too far out of state, and where it gets you isn't any place I'd consider a family vacation destination.
Some of my favorite memories as a kid were of West Texas, which has already been suggested. While sparse traffic is a plus, it can also be a negative, as people aren't accustomed to looking out for cyclists. The parents of one of our daughter's friends go to Alpine every year. He is an avid cyclist and always takes his bike. To my knowledge, he always comes back in one piece. Thus at least parts of West Texas are doable.
Austin actually has a lot to do for the entire family, and it is by far the most bike friendly city in the state. It has several designated bike lanes throughout the city, and many undesignated roads that get a lot of cycisits, so drivers know to be on the lookout. There are many places to ride offroad as well if that's an option for you. It also isn't going to cost you a fortune in gas in order to get there.
We typically take the family to Port Aransas. It's only a couple of hours away by car. While I've seen many a crusty local on a cruiser, I can't say that I've seen many serious riders. I've seen a few brave the stretch of 361 between Corpus and Port Aransas, but I'm not sure I'd do it. While the shoulders are ample, the traffic whizzing by at 70+ doesn't exactly make for a relaxing spin. If you frequently ride along 1604 or 10, you are probably acclimated to riding along side fast moving vehicles, so it may not bother you. There are also a lot of smaller highways that would get you out into the coatal plains toward Sinton, Taft, etc.
I would imagine that South Padre would be an option as well, but I haven't been in ages, and I have no idea how safe it would be for cycling.
If you end up going to one of the coast options, you'll want to clean your bike after every ride and keep it inside. The salt air can toast your components very quickly.