Vehicle Registration & License Question

I'm moving this weekend to Texas, and have a DMV appointment set up on Monday morning for a Texas license. My current state's license expires on Tuesday, so I initially thought this was all perfect timing.

However, I'm now seeing that I need to provide proof that I've registered my vehicle in Texas before getting a license or sign saying I don't own a car.

Has anyone signed saying they don't own a car (obviously when not true) and had any issues? I plan on registering my vehicle in Texas this week, but would rather get a license before I have to deal with repeating the driving test and all due to an expired license.

Going back to my home state to get a new unexpired license is also not possible, unfortunately.