I'm having a dry cough, and runny nose and it feels like my chest is congested. Is this a common cold? Will OTC medicine work?

Common Cold. If your dry cough persists for a while (4-6 weeks) this could be secondary to inflammation in your upper airway or lungs subsequent to the common cold viruses or post nasal drip syndrome, in both these cases a short course of inhaled broncho dilators like albuterol might be useful (this is not OTC). Try steam inhalation, it provides symptomatic relief.
Sure. It sounds that you have a common cold. Watch out for fevers though. I believe that OTC medications can help alleviate some symptoms. Make sure to take them as indicated, do not over use them, drink plenty of fluids, and rest. Your body will recover soon.