I'm all good has more contexts, because it can apply to situations where someone asks you if you're OK after something happened, ie, 'Are you OK after the fall? - yep, i'm all good'. I'm all better usually refers to illnesses.
I'm is present tense, so because it was last week, you'd say instead, 'I was coughing seriously last week' or 'I was coughing severely last week'. By convention, people tend to say 'pretty seriously' or 'pretty severely' - no reason, just convention. Hope that helps!