The tricky part is the quiet part. As you might have read on some of the other posts, state parks are still busy on the weekends. You might have better luck finding seclusion in national forests and grasslands.
http://www.fs.fed.us/r8/texas/ I have never camp in them before so maybe someone else can post some information.
For nature, I like to head to Caddo Lake State Park. It's a whole other world from metroplex. It's three hours east of Dallas. A must is a patoon boatride through the bayou. There is a guide who operates out of the concessions in the park. $10.00 per person the last time I went. Worth every penny. The guide was very knowledgeable of everything that slithers, swims or flys in the bayou. Dont' forget your binnaculars. Call the park headquarters to get the telephone of the consessions store for trips.
Not as far, is the Tyler State Park. Very beautiful and has lots of tent sites.
Motorman355