请帮我看下我写的句子有错吗

你的意思能明白,但这个句子还可以表达地更好点儿.
Due to 用在这里不太合适
responsibility是不可数的,不能跟a

建议修改为下面的句子好一点儿
Seeing his father getting older and older, Jack eventually realized that he need to carry on the responsibility for his family.
眼看着他的父亲一天天变老了,杰克逐渐意识到他需要承担起家庭的责任了.
Having noticed his father is getting older and older, Jack realizes it is time for him to carry on the responsibility for his family.
注意到他的父亲一天天变老了,杰克意识到是时候该他承担起家庭的责任了.

希望我的回答可以帮到你.

[地道英语兴趣组]