I feel bad is correct. In this case, "bad" is what is known as a "predicate adjective". Predicate adjectives follow certain types of verbs such as: seem, appear, look (he looks bad), sound, feel...
You feel a dysphoric feeling - meaning "not so well feeling". Your body seeks warmth, so as to avoid unpleasant skin sensations, joint stiffness, runny nose, and concentration which is sometimes not...