제 입은 무거워요.

"My lips are sealed"

오직 다른사람의 개인적인 이야기에만요.

But only about other people's personal matters.

그러나 제 이야기에서는 제 입은 무겁지 않아요.

"About myself, I will talk freely"/ "I talk freely about myself."

제 생각에 전 제 이야기를 할 때 주의해야 할것 같아요.

I think I should take more care when I talk about myself.

Notes: I made the translation more natural.

1. "이야기" is story, but in English we would say "matters"

2. "입은 무겁지 않아요": there are actually a lot of ways we would say this, but we normally wouldn't say "my lips are not sealed".

3. "주의해" This is normally translated into "be careful" or "take care".  You don't have "more" in your sentence, but in english, the grammar demands "more". Caution is the noun form of it.

If you have any questions about my changes let me know.