For a new (Texas LTC) applicant a new set of prints will be required. Even if the prints had been done in Texas for another requirement (ie, RN, EMT, real estate agent, adoption, etc), a new set of prints would be required (they're only used for the sole purpose of that specific background check).
Once the DPS has a good set of prints for the LTC, they will continue to use those for all future renewals (unless you let your LTC expire for more than 1 year, & the DPS purges your data from their system. In that case, you'd need to re-apply as if you never had a TX LTC).
But, as Jusme said, it's a relatively easy process.