이 세상 산림엔 꿈도 많고
우리네 가슴엔 꿈도 많네.
Here is the some personal suggestion. The meaning is exactly right, and any korean understands. but , sometimes you don't need to use "의" like "이 세상의 산림엔", rather "이 세상 산림엔" this is more musical. and the end of the poem that you wrote, "많다" is more declarative than "많네", so I suggest "많네", there is little bit difference. I hopefully you like, and wish to see your drawings~
Good luck