Hi, in my opinion flu
symptoms take almost takes a month to go but , as you say the black spots in phelgm is not a good sign. Needs to evaluated.
If I were your treating doctori would asked for blood work, sputum analysis, chest xray to rule out the cause.
Depending on thecreoort I would have treated you. But I would advise you to continue taking anti inflammatory and decongestant medication your symptoms of flu.
I woukd advise you to meet a physician and grt yourself treated.