If there's time to get the passport back before you fly then do that and book under your legal name. But i think as long as your ticket and passport match you should be OK. I hadn't changed my passport after I changed my name, so when I had to do a last-minute overseas trip I booked it under my maiden name. I brought my marriage licence along just in case but no one ever asked me any questions about it.
I am booking airfare but I need to renew my passport before I leave. I also had a name change since my last passport. Should I book travel in my new name and use that when I apply? I'm confused about how this works.