I almost laughed out loud when I saw the topic of this post, because it is pretty much the story of my life. Phone interviews, especially, are terribly fertile breeding grounds for awkwardness. Something I've learned specifically for phone interviews is to allow extra time for the person to talk- maybe it's just me, but when I'm talking to people on the phone I can never tell if they're done talking, and I think they are done, then I say something, and they're saying something, and then we both stop talking, then we both start talking again... not good. I've learned to pace phone conversations differently than real life ones.
However, one advantage of phone interviews is that they can't see you! Feel free to write notes and have them in front of you- questions to ask, research ideas, words of encouragement, etc.
Also, don't be afraid of small talk at the beginning- things like how they got started in the field or whatever. That might put you more at ease.
And yes, the professor might be awkward too! In fact, they may be more awkward than you! That's the nice thing about going into academia.
Sorry if these are obvious to everyone else, because they are things I had to learn from trial and error Good luck!