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what will be the best word in korean saying " i want to share the rest of my life with you?"

The simple expression is "나는 너와 인생을 같이 살고 싶어."

I'd say that "너와 함께 남은 여생을 함께하고 싶어".

1) I want to ~ : ~하고 싶다.
2) the rest of my life : (나의) 남은 여생
3) with you : 너와 함께
4) I want to share something : ~함께 하다 or ~함께 공유하다.

To sum up again, "(나는) 너와 함께 (나의) 남은여생을 함께 하고 싶어".
^___^

This expressin is usually used for proposal for marriage.
It can be translated into Korean like this "당신과 함께 살고 싶어 (나와 결혼해줘)".

I hear 여생 said by someone middle-aged or when talking about a middle-aged person.

I suggest you say 너랑 평생 함께 하고 싶어.