It is I that who am unwilling to come .

原句错误!后面的who是多于的!可改为下面形式的强调
It is I that/who am unwilling to come .
句意：是我不愿意来的!【而不是别人】
类似句式有：It is I that/who am to blame.
受责罚的应该是我!
把that/who去掉,句子“I am unwilling to come”是正确的,不过就是普通句,无强调之意了.翻译成“我不想来的”
“总觉得who am 挺奇怪的”,可能你对这样的用法还不熟悉,所以觉得怪怪的,其实它是正确的!
多看几眼就习惯了,学英语需要积累的.
Wish you success!

It's me that unwilling to come

读起来怪怪的。
应该用It is me that who is unwilling to come .
that who 最好不要去掉啦，两个都去掉后肯定不对的