It seems like postnasal drip in which excess mucus trickle down from the back of the nose and continously irritate the throat thus causing constant cough
Stay hydrated to make mucus thinner.Drink 8-10 glasses of water /day.
Salt water gargles are useful.
losenzes are to be chewed thrice daily.
Allergens should be avoided as they increase mucus production like dust,chemical sprays and even humidity.
Neti will wash the passage and mucus but should be done under supervision of ayurveda expert.