Changing your name and gender marker on your driver’s license is NOT the same as a legal name and gender change. A court order for name change is much stronger than a driver’s license or US state ID. It will be important for changing other identity documents.
For US citizens
You need to get your legal name change in court. Failure to do so may mean you are not legally considered your new name, which can lead to legal problems involving employment, marriage and other matters down the road.
With the passing of the REAL ID Act, US states now link their driving records with Social Security records, passports, and other federal documents. You may need to get your Social Security information change before your driver’s license in some states. You may also need to get your birth certificate change (where available) based on your state’s requirements.
National Center for Transgender Equality (transequality.org)
Disclaimer: This is legal talk, not legal advice. Laws vary by jurisdiction and change often. Some of this may not apply to you. It is presented without warranty. It may contain errors or omissions. You must do your own research.