From looking at the DPS website, it seems as though because there is no reciprocity agreement between the U.K. and the US, in order to get a Texas licence, you won’t be asked to give up your U.K. licence anyway.
It just says that to get a Texas licence you have to take and pass the Texas driving tests. There is no mention of having to give up your U.K. licence at all.
And even if your licence was from a country with a reciprocity agreement, if you didn’t want to give up your foreign licence, you would simply have to take the driving tests instead of exchanging your licence.
http://www.dps.texas.gov/DriverLicense/movingtotexas.htmFrom that link:
Applicants who have a driver license from any country other than the U.S., Canada, France, South Korea, Germany or Taiwan must take and pass both the knowledge and skills exams.
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