Yes - definitely you need to be checked out by your doctor or an ENT specialist
if allergy testing has shown that you are not sensitized to aeroallergens.
If an extensive allergy panel was not done, it would be helpful to ensure you do not have mould allergy
or a condition called allergic fungal sinusitis
that can be difficult to treat.
Post nasal drip does occur in non-allergic rhinitis
and causes the dry irritating cough so nasal corticosteroids
long term use offers the best solution.
Try and avoid very strong perfumes, or chilli that aggravates the rhinitis.