나 오늘 몸이 별로 안 좋아요. Two subjects appear in one sentence in Korean. When you strike up a conversation, you can say 나 오늘 몸이 별로 안 좋아요. If someone asks you to go out with him/herself, it'd be better to omit "나" since s/he knows it's you who feels sick.
It'd be probably helpful to see the structure. 오늘, 나는 아파 느낌 > Today, I'm sick. feeling.
오늘 메스꺼워요 > Today it's queasy. (The subject is not obvious here.) "오늘 속이 메스꺼워요." would be what you actually meant. I usually say 오늘 속이 안 좋아요.
Be careful not to use 은/는 very often. 은/는 is used to seperate onesef from others. Use 은/는 to describe the characteristics of the subject(like 저는 한국 사람이에요), to say something opposite to what someone else has said(전 그렇게 생각 안 해요.) If you use 은/는 without being aware of it's use, the listener could feel isolated.
I don't think "I feel..." exists in Korean. English expressions sometimes make sense only in English.