This is wrong. Start with SSA. Your marriage license is just the proof that you meet SSA’s requirements to change your name. It doesn’t need to have your new name. When you applied for, and signed the license, you still legally held your maiden name.

Editing to add, when I got married I paid for one of those name change companies off Groupon. It’s actually very useful because they have aggregated all the forms for major companies and governments, and you select the ones you need. It then works as a template where you fill out your info once and it prints it on everything for you. And it produces a checklist for the requirements of each company or state government. Well worth the $40 I paid!