Hello
You are having "
dry cough "  from last many months usually worse early in the morning (as mentioned by you in your query ) . 
This type of cough ( dry ) may be due to these reasons , these include.
1 Smokers cough , if you are smokers then this may be a prominent reason of such cough . Dry cough present because of allergy to smoke.
If you are not a smoker , inspite of this you may have a dry hacking cough due to allergy to certain allergens, these are  dust , mite , pollen , mold , sudden increase or decrease in temperature , humidity , fine dusting powder.
Such allergen causes 
bronchoconstriction  , hence 
bronchospasm , hence tickle in throat or lungs and cough . 
When such patient visit I prescribe them to take monteleukast+ 
fexofenadine tablet once or twice in day . Also take terbutalin+ guiaphesine + dextromethophan syrup . 
2 As I mentioned above 2nd reason of such symptoms may be due to astmatic allergic 
bronchitis . All  allergens mentioned  above may be the predisposing factors of  Asthmatic allergic bronchitis ( usually in this type of dry cough there also develops breathlessness i.e. shortness in breath ) I usually prescribe monteleukast= bambuterole tablet once / day.
3 Sometimes foreign particle inhalation causes such sever dry hacking cough . So if you  think that something is inhaled  
consult a pulmonologist and get 
bronchoscopy for removal of inhaled particle .
Hope this will help you.