The Korean verb closest to "realize" is probably 깨닫다, but 깨닫다 is much more serious sounding than "realize". I doubt anyone will say 깨달았어요 in the situations of your examples in real life. We typically use more casual phrases in such contexts. - I just realized that I don't know your name! => 그러고 보니 아직 (서로) 이름도 모르네요. - Wow I just realized what time it is! => 이런, 벌써 시간이 이렇게 됐네. (벌써 has the sense of realizing it belatedly). These sentences don't match the English originals exactly because there's no verb corresponding to "realize", but this is how we say it in Korean. The -네 ending is natural here because it expresses a light exclamation of surprise. - 그러고 보니 ... -네(요) / 생각해 보니 ... -네(요) => Come to think of it, I ...; I just realized ... - 알고 보니 ... -네(요) => I didn't know but ...; I just realized ... Other examples: - 그러고 보니 은행에 들리는 걸 잊어버렸네: I just realized that I forgot to stop by the bank. - 알고 보니 우린 같은 학교를 나왔어요: I didn't know (before) but we both went to the same school. For more serious things, we use 깨달았다. - 그것이 나의 잘못임을 깨달았다: I realized it was my fault. - 직장 생활을 해 보고는 세상이 만만치 않다는 걸 깨달았다: I realized the world is no picnic after a taste of the work life.

I just realized = 지금 막 깨달았어 / 방금 깨달았어 "I just realized that I don't think I know your name!" = 내가 당신 이름을 모른다는 걸 방금 깨달았어요 "Wow I just realized what time it is!" = 지금 시간이 이렇게 된 걸 방금 깨달았어요.