I'm not very sure how you pronounce your name. If the first A of Adan is the same as A of April, we write エイダン. If it's as same as A of "actually", we write アダン.

Yeah, I think that's right. I also believe if you write or type your name it has to be in katakana. Therefore it will be "私はアダンです。"

Yes. Saying " watashi wa Adan desu" means "I am Adan".