Cough discharge. For permanent solution?

Since 2 weeks I was having cold and cough, for which I took medications from general physician. I am better now, but there is this constant sensation of cough in my mouth near the throat, and when I cough it out, small quantity of cough comes out and its been there since 3 weeks now. I have taken medications, cough syrup, zinda talismath, but nothing has stopped it. Though its not totally irritating but it is taking a silent toll on me. Since last 2 years my cold and cough pattern has changed. The nose rarely gets blocked, it just keeps coming from the mouth. Just a little back ground, I get recurring cold every 3 to 4 months, with any consumption of cold things. Kindly help me on this. Should i turn to homeopathy for permanent solution? Look forward to an reply at the earliest. I am healthy person weighing 96 kilos with good physique and healthy eating habits. No drinking or smoking. Have a history of cold and cough in my family right from my dad's side of the same.