英语,有语病否?many students graduate unaware of the best methods o

正确,没有语法错误.
1 unaware of the best methods of searching for a job可以说是many students的后置定语,现在放到谓语动词的后面了,目的是避免头重脚轻.
2 unaware of the best methods of searching for a job也可以说是主语补足语,用英语句子表达就是：Many students are unaware of the best methods of searching for a job when they graduate.
3 再补充一句有主语补足语的句子：
He often goes to bed hungry ==He is often hungry when he goes to bed.

可以，没有问题，这里相当于
unaware of the best methods of searching for a job 这部分做了句子的伴随状语，
通常我们会用现在分词做伴随状语，可以在整句前加一个being，成为
being unaware of the best methods of searching for a job

没有语病哦

the sentence construction is not between graduate/unaware, it is in the way you put words together.
How about: Many graduate students unaware of the best method of searching jobs. If you use the ...