I have been having both a wet and dry cough for 3 weeks but when I cough I find it hard to breathe and my lips turn a bluish color, what does it mean?

It means see doctor! If you are coughing enough that you are becoming short of breath, the cough is productive, and you've already tried medications (you list mucinex/robitussin/ and antibiotic Amoxicillin in your med list), then you need to see the doctor. Blue lips can mean poor oxygenation and i'm worried that you say you find it hard to breathe; you really should see someone today, so consider urgent care too.