I agree with
krazykanuck on this.
I've only lived in Texas for 1-1/2 year, and am by no means an expert. I had visited quite a few times before that...but it really is such a big state! Even the areas east of the I-35.
I'd try to fly in/out of DFW and IAH, then piece together a trip between them (reverse if you do the opposite):
* Fly into DFW, spend 3-4 days exploring that area
* Rent/hire a car and drive down to Austin, for 2-3 days.
* Drive down to San Antonio, for another 2-3 days
* If you want the beach, go to Corpus Christi, 2-3 days
* Finish up in Houston with your remaining time, fly out of IAH
I've only been to Corpus Christi once, and it was a while ago, but I really enjoyed it (we're going in October, so perhaps I'll remember to check back then).
I remember driving down the length of Padre Island to the National Park, and that was probably the best part.
There are beaches further down, South Padre near Harlington, but they tend to get more college kids...or they did when I was in college at least.
If you're looking for some non-city options, I recommend watching this video:
http://texaspbs.org/nationalparks/
The Texas PBS stations collectively put it together, and it should give some great ideas.
Additionally, I'd also check out some episodes of
The Texas Daytripper, which also comes on our PBS stations. For example if you do more time in Austin, you can take some daytrips out to Marble Falls, Fredricksburg, etc. You could even stay out in some of those smaller towns/cities.
In addition to Texas, you could consider visiting some nearby areas, such as New Orleans or even Oklahoma City. They would give quite a different perspective on the area (and I could probably guarantee you wouldn't see any other Brits in OK City).
We went to Dallas last fall, and did many of the tourist sites:
* Perot Museum (would be great for kids)
* Reunion Tower
* 6th Floor Museum, plus walking around Dealy Plaza
* Arboretium
* Plus ate at many great restaurants
We got the City Pass, which included many of those. I'd still like to see Southfork Ranch, and maybe the Bush Library, and spend some time exploring Deep Elllum.
(by the way, if you stop in Austin, definitely visit the LBJ Library).