Vasomotor Rhinitis. Your symptoms are consistent with a condition known as vasomotor rhinitis. This is not related to infection or allergies. Generally, this is easily treated with a prescription nasal spray spray called atrovent (ipratroprium bromide). See your primary doctor or ENT for full evaluation.
Neural response. This phenomenon is usually not allergy. It is most commonly the result of a central triggering of the vagus nerve. This nerve innervates both the blood vessels and the mucus glands of the nasal mucosa. It is possible that an anti-cholinergic nasal spray might control the rhinnorhea (runny nose). It is a perfectly benign condition.