In Texas, I think it is better to change the status. We have never been able to update my wife's address online. I assumed it is because she was a Green Card Holder. I, as a USC, have always been able to update my address online. For that reason, I plan to get wife an appointment at DPS to update her status to USC.
I am going to try online again, but the DPS system is undergoing Maintenace at the moment. I did find that even to change the Texas DL address by mail, the person MUST be a US citizen. I also found the same requirement exists for online address changes.