Child has dry cough. Sinarest okay? Or take Augmentin?

My 2yr 10 month old son has been having dry/ slightly wet (for a want of a better word since i can hear a slight phlem build up) cough since the last two days. at night he s almost coughing continuously. last evening he has temperature for a short while but crocin took care of that. On 27 August (3 weeks back), we d taken him to the doctor for flowing nose and temperature - wherein the doctor prescribed sinarest and also gave us Syp Augmentin DDS for use if it worsened. Since the last two days, we ve been giving him cough syrup Zedex (2.5 ml each morn, noon and night) but since it did not seem to help, gave one 3 ml dose of Syp Augmentin DDS today morn. Googling i understand this is for throat infection. my question: 1. Is it ok to continue the Augumentin dosage for the next 7 days or should i discontinue as i ve given only one dose? 2. what would you suggest for his cough? Thanks in advance for a detailed reply.