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!

you can say : I had a heavy/bad/severe cough last week. I feel good./ I feel much better.(You should use"feel",not "am",because if you say: I am good,means I am fine,I am ok,not the same meaning)

I would also suggest replacing or moving the word seriously. I have heard of having a serious cough. This is a cough that may be of a health concern. This phrase seems to have always been "a serious cough." Never "coughing seriously' or 'seriously coughing." To change the word with ly would make you a sincere cougher. Though I do not know you well, my guess is you are sometimes serious/sincere, but rarely would that show through in the way you cough. Between you and the cough, i bet the cough was the serious issue. I guess that is splitting hairs. So- I might suggest these too options. I was coughing severely last week. I had a serious cough last week.

I WAS coughing seriously last week.

I was coughing