Hmm...allant...I looked it up, it's the present participle of aller. I don't recall ever learning about the present participle of any verb before.
Presumably, the point of the lesson is to teach the "while [do]ing" construct.
Je suis tombé en mangeant de la soupe. I fell down while eating soup.
That phrase gives confusing context. Does "se sentir mal" mean physically ill or psychologically or both?
It can be literal or figurative. Je me sentais mal could mean I used to feel bad, sick, anxious, worried or simply "I have a bad feeling" (à la Star Wars ☺).
Whatever the context behind the emotion, it is safe to translate mal to bad.
I wrote "Every day I used to feel bad on my way to high school" What is wrong with that? It is much better English.
en allant = "while going", not "on my way"