Birth certificate
From Texas Name And Gender Marker Change
Lambda Legal's has a guide on changing birth certificate sex designation, by state. Four states do not allow amending sex designation on birth certificate: TN, OH, KS, ID.
National Center For Transgender Equality also has a guide on changing birth certificate sex designation, for all states and U.S. territories.
Guam
Guam requires GCS to amend sex designation on birth certificate.
Illinois
Name change on birth certificate: Download a form, fill it out, send it and court order for name change. They'll send everything back with the corrected new birth certificate.
Puerto Rico
Puerto Rico does not allow for changing a gender marker on a birth certificate.