You use "sentir" instead of "sentir-se" when you have the reason why. Ele sente medo do escuro = he is afraid of darkness. In this case, he doesnt feel "himself" well, not because of an external factor. Then, as it happens tp himself, use the reflexive verb "sentir-se"
I like your explanation of 'sentir-se'. A literal translation of 'sentir' may help make the difference between them even clearer.
Ele sente medo do escuro = (Literally) He feels fear of the darkness
It's one of those cases where English speakers need to know the literal translation to help them remember that Portuguese uses a different verb to get the meaning across.
In English, it is understood that this sentence is about his health. In Portuguese, and in Spanish, it most be specified that the feeling is about himself. "Ele não sente bem" without specifying that this is is about himself could imply that there is something wrong with his sense of touch: can't feel other objects very well.