As someone who is a Canadian but living and working in the US prior to the start of my transition and thus far, during transition, I don't think you'd have much to worry about (although certainly different culture and such). My passport still has my original birth name and gender (due to reasons beyond my control, I don't think I'll be able to get gender officially change on my passport until much, much later and after more than just top surgery). That said, when you did your name change you were likely given a piece of paper signed by a judge about the name change.
I've been told that as long as you have that, along with old/new documentation you should be fine. Most jobs elsewhere (outside your home country) will just need a passport since it's considered to be a valid, recognized identification document. You could show them the passport and name change paper, if the new passport hasn't arrived and explain the situation. I would imagine that most customs agents likely won't care for the reasons but as long as it's documented, that's all that will matter.
