I applied for a passport and passport card first. Once in hand, I went to the SSA office with my passport / certificate (when I got my green card, I also went in and got an unrestricted card at that time) however I also wanted the SS office to be updated showing me as a US citizen.
"This database is shared with all federal agencies and many state agencies further validating your US citizenship status with the federal and state governments. With your social security records updated showing you as a US citizen, it will be much easier in the future to apply for and receive all social security benefits entitled to you".
Subsequent to updating the SS that you are now a citizen in their system, if you happen to lose your unrestricted card in the future, you can also apply for a replacement SS card online .
Take passport to DPS and have them change their files - the benefit of changing status at the DPS is that the next time you need to renew your license, it can be done on line. Take your unrestricted SS card with you - just in case they want to scan it for their files. I don't believe I took my NC with me.
https://www.votetexas.gov/register-to-vote/update-voter-registration.html