Are awkward phone and in person conversations with new significant others normal?

Yes. Every relationship is different. Sometimes two people who like each other are nervous, because they are concerned about making a good impression and are afraid of loosing the other person. Sometimes two people click instantly and feel like they have known each other all their lives. The best thing for you to do is just be yourself and encourage your significant other to do the same. That way, you can build a relationship built on a foundation of who you really are. If you are constantly worrying about trying to impress the other person, you may not be right for each other.