Your symptoms are suggestive of post nasal drip. It is also called as Chronic Upper Airway Cough Syndrome Secondary to Rhinosinus Diseases (Postnasal Drip Syndrome). The pathogenesis of this cough is mechanical stimulation of the cough reflex in the upper airway. The clinical presentation are combination of any of the following;
1) a sensation of something draining into the throat,
2) a need to clear the throat,
3) a tickle in the throat,
The treatment is;
1) nasal saline spray; do not need prescription for it.
2) nasal corticosteroids / nasal cromolyn / nasal antihistamines may also be needed,
3) oral leukotriene inhibitors and
4) oral antihistamines; you may try Claritin D which is over the counter.
You may consult nurse practioner, who can provide you the prescription for these medicines.
Please feel free for your follow up questions.
I would be happy to assist you further, if you need any more information.