有没有feel like that.这种说法如I feel like that I am chosen.

有的,feel like 有不同的意思, 表示 喜欢, 想要, 感到, 意欲

有. Very common.
For example:
I feel like a millionaire.
I feel like a thousand bucks.

feel like 一般后面加doing

I have a feeling that I am chosen.

没有,除非 that 后面不接从句,在前文交代了从句的内容,然后在这里用 that 代替从句.
而且 feel like 后面一般加 doing sth.,比如,I feel like slapping him.

有啊 i feel like we have to punish him ,. 牛津教材 1模块2单元 reading 部分