I had asked for some further information and have not heard back.
There are several possibilities of how a cold can progress to causing chest symptoms such as you describe.
The most common situation is that a viral upper respiratory tract infection will directly spread to causing a viral bronchitis, an infection of the airways of the lungs, which can cause significant coughing. It is also possible for a viral infection to cause inflammation of the lining around the lung, which is called pleuritis or pleurisy.
However, another possibility that would be more of a concern would be the development of pneumonia, an infection of the lung itself. If you are having significant shortness of breath, then that would be more of a concern. If you are having significant shortness of breath, then it would be important for you to be seen by a doctor soon.
If there is no shortness of breath, then it would be reasonable to continue managing this at home. A cough syrup can help to ease chest symptoms, such as guaifenesin plus dextromethorphan. For the pain and shock sensation, it also may help to take an over the counter anti-inflammatory medicine, such as naproxen.
If I can provide any additional information, please let me know.