You can generally go get your Texas driver’s license basically immediately. You need to make sure you have the right documents (passport, proof of SSN, and your two proofs of residential address).
Two Proofs of Residential Address: You must bring two items with you from this list. Typically, the most convenient items are a cell phone bill, pay stub, or bank statement (dated within the last 60 days). The address on this document should match your residential address as listed on your Address Tab. This item does not need to be an actual piece of mail but it should be printed.
Texas DPS says:
Name and address changes must be updated on BOTH your driver's license/ID Card and title/registration within 30 days of changing your name or address.
Although you can obtain your driver's license later, we recommend doing it as soon as possible to reduce risk and eliminate confusion with your previous state of residency.