In that case, if I just want keep two different names as it is now, how will that impact my travel?

You leave the US and come back to it on your US documents and you travel the world either on our Reentry Permit or your country's passport. When you enter you country you use that passport.

There is nothing illegal in using two names in this specific set of circumstances.

Good luck. By the way, my answer was the best, I hope you mark it so.

Thank you very much for your help :)