No, "I'm being sick" is you are doing something, literally being sick.
"Sono" is to be something, try to think of it as what you are, (I am a human being).
"Sono stanco". I am tired. You're not doing tired but being tired.
When you say "sono stanco" you are that tired person (human being).
That's how I get my head round it.
Sometimes you just need to memerise the sayings.
Sto male. I'm sick/ill. Sto bene. I'm fine/well.
There are a lot of other phrases, I am cold, I am afraid, I am hungry
You don't say "sono freddo" (I am cold) but "Ho freddo" (I have cold) from the verb avere. Ho paura (I'm afraid). I have fear. Ho fame (I'm hungry). I have hunger.
All very confusing! Memerising the phrases I think is the best way.
I hope this helps.
I'm sure Maria can clarify things better and point out my mistakes.