What Is The Cause And Treatment For Cold And Cough In An Infant?

My son is 1 yr 9 months old. We moved to Toronto, Canada last week.
From the second day of our arrival in Toronto, he has running nose, and was coughing intermittently. He was never exposed to the weather outside (except few minutes to get in/out to/from taxi etc when we arrived here). We initially thought it would be okay in two three days, however, it did not.
He was given Ambrolite D cough syrup twice a day and that reduced coughing to an extent. But whenever he coughs, it sounds like he has some mucus left inside. Cough does not have any blood spot.
He hasn't had a fever during this period. Food intake is little less than normal, stool and urine are as usual.
Sometimes his nose gets dried up and becomes congested. I consulted a walk-in drug shop here for a nasal drop and they have suggested 'hydrasense' (supposedly made with 100% undiluted natural-source seawater), and we have used that twice a day for last three days.
We have stopped bathing him, instead sponged with hot water.
It's been 7 days now. Would highly appreciate if you could advice.