Hello and hope you are doing well.
The other causes for long term cough besides asthma are GERD and post nasal drip from allergies. GERD is gastro esophageal reflux disease where the stomach acids come back into the esophagus and spill into the wind pipe causing cough. Treatment with antacids will help neutralize the acids. Post nasal drips can occur with sinusitis and throat infections. Giving up cigarettes will abolish smoker's cough in 90% of people. Talk to your GP about the other causes for cough in the meantime you could try warm salt water gargles and breathing exercises.
Hope this helped and do keep us posted.
Was a sputum sample or a bronchoscopy done? When these are done, a cluture is done to identify actual organisms that are causing your infection. Calrythymiacin is used to treat Neutrophil Asthma caused by both clamydia and mycoplasma bacteria. The chronic mycoplasma infection is easier to treat than the clamydia infection. If it is a clamydia infection it will take a while for the antibiotics to work.
God bless.
Did they test to identify the organism? If not, and the organism is a fungus, the antibiotic just made the infection worse.