The answer is YES, there are definitely homes in the Dallas/Ft. Worth area that have a problem with Radon and they don't need to have a basement for an issue to occur. There are several homes I have personally visited that have a mitigation solution in place. One prospective home buyer turned it off (guess he didn't believe in an invisible gas) and then had it tested. Not surprisingly, the levels went well past the safety limit until the system was re-engaged and left on.
While it is not as bad a problem as it is in many states in the Northern part of the U.S., the % of homes in the DFW Metroplex is substantial enough to warrant testing. There is a breakdown by county here: http://scantech7.com/radon_faq_dallas_ft_worth.htm but the short answer is that between 5-10 % of all homes in North Texas exceed the EPA action limit of 4 picoCuries/per Liter.
And you can get the tests yourself at Lowes (maybe Home Depot) but there is a significant time factor to keep in mind. The tests require at least 48 hours to develop, but you cannot leave them out for longer than 96 hours so that gives you a 2-4 day window in which they must be picked up. Then you seal them in a special envelope and mail them in. The results can take a week or more unless you pay a rush fee and even then it could still be a few days on top of the minimum 48 hour deployment.
It is more economical to handle it yourself, but it is also a hassle and easy to do the test incorrectly by improper placement or forgetting to put the time stamp / kit code on the paperwork. I have even seen experienced home inspectors and other paid specialists (no offense) accidentally invalidate the test.
If you are in a rush and want it done right, then consider hiring a professional.
Joel-Anthony Gray of ScanTech Technical Consulting