You did not mention if you have any other symptoms of nose other than post nasal space.
With the information provided by you it appears that your post nasal drip
and bouts of dry cough are due to allergy.
But similar symptoms can be there with sinus infection
also. So it is important to examine you to differentiate between the two.
For this please see an ENT specialist
who can advise you regarding further management.
If it is allergy alone he may advise you antihistamines and steroid nasal spray.
Till the time you see him/her do the following things:
Avoid coughing strongly and try to hold coughing as much as possible but when you can not hold any more cough gently. Try taking deep breath in between.
You can do steam inhalation if it helps.
It is doubtful whether you have asthma if there is no response to bronchodilator
. However if the wheezing persists you can consult a physician (Internist) or pulmonologist.
does not relieve your post nasal drip.