Just make the names and addresses of every couple getting a marriage license public record. Put it in the newspaper.
Most places they do. And even if they do not put them in a newspaper, they are matters of public record. That is kind of why she has absolutely no basis to refuse to issue them, any more than she can refuse to do the other parts of her job. She works for the public and cannot decide that some laws simply will not apply.
Why not? Obama has being doing that for 7 years now.
No, he has not. He has not refused to enforce a single law. But, I am sure you believe otherwise. And, when a Judge in Texas ruled that the administration's view of its authority regarding the law was incorrect, the Administration complied with that ruling. If some think he is not following the law, they need to take him to court. If they win, he will certainly abide by any court ruling. Until there is a court ruling that the administration is not obeying the law, the law is being followed.