OK. I would say a couple things:
First, if you have had symptoms that have not gotten better for a month, that would merit a bit more investigations, starting with a chest x-ray, to check for pneumonia and any complications (eg. fluid around your lung) and some basic blood work like a blood count.
Also, in the United States between 1/4 to 1/3 of the most common bacteria that causes a bacterial pneumonia is now resistant to amoxicillin, so while it is an OK first line choice of antibiotic, there are certain people who will not respond and need other antibiotics.
Having said that, the most likely explanation is you have a post-infectious cough- a form of airway hyperactivity that continues after an infection. This is like a form of asthma that comes on with an infection and carries on. I often treat this with inhalers - inhaled steroids and "bronchodilators"- puffers like "ventolin". You also may benefit from some anti-histamines or decongestants for your other airway symptoms.
I would recommend you follow up with your doctor (or an infectious diseases doctor) for these tests and recommendations.
I hope that helps. Always ask if you need clarification/more information.