Did this recently. Walk in to SSA with a completed social security card application (https://www.ssa.gov/forms/ss-5.pdf). There’s space on the form to include what name you’d like as well as what was listed on your former card. Bring your marriage certificate and passport. The DC certificate that only lists your maiden name is fine as long as the name your changing to is some combination of your name and your spouse’s. Once you get the confirmation from SSA, you can go change your license at DMV. Then everything else. It was actually all easier than I expected. Good luck!