If I can't do it on time, which name will I put for my airline ticket? Thanks for your advice

It is my opinion that you should not travel without changing your name on the passport. They are not going to just believe that you are the same person because you say so and it will probably lead to issues at the border. Passport applications can be expedited and you should start on the process right away.

And the name on your ticket should reflect your current legal name.