Getting rid of a chronic cough is a difficult problem. It is also fairly common. If you are continuing to have a cough or feeling shortness of breath I strongly encourage you to see your doctor
Firstly, making the diagnosis of what is causing your cough is important. Viruses can cause a cough. This can be in the acute setting such as the common cold or upper respiratory infection (URI). People can also get post-viral bronchitis--which means the inflammation lasts after the virus and therefore the cough continues. Unfortunately, there is no actual treatment for this. Antibiotics work to kill bacteria, not viruses. In fact antibiotics can worsen some viral infections. If you are confident this is from a virus then antibiotics are not appropriate. There are other medicines that can help symptoms...talk to your doctor.
There are some conditions that require antibiotics. Pertussis or whooping cough can cause a long lasting cough that can be treated with antibiotics. So can pneumonias and they require urgent antibiotics.
Unfortunately, diagnosing what causes the cough is hard. There are many causes of chronic cough. Asthma
is a common one that often does not present with wheezing as in most people, but just a cough. Acid reflux
can also cause a cough commonly. Allergies or post nasal drip can also cause a cough. Smoking or other irritants also cause chronic cough.
I encourage you to talk to your doctor to make the diagnosis as well as start treatment if appropriate.