Upvote:8
You should book your flights under the same name as the passport you will be using to board the flight. If you have a new passport with your new name, then you should use your new name to book your tickets.
We've been in this exact same situation before: when we got married, my wife had her USA B1/B2 visa in her passport with her maiden name in it. Later, she obtained a new passport with her new name on it - and the old passport was cancelled and returned to her. The next time we travelled to the USA, she used her new passport (with her new name) to book tickets - and took her old passport and our marriage certificate with her.
When entering the USA, she presented all three things to the border agent - and got through without any issues.