Hello
Welcome in H.C.M.
I have gone through your query regarding tickly feeling of throat along with productive cough. As you said that on lying down your one 
nostril gets blocked and you feel better in sitting posture it indicates that you are suffering from upper 
respiratory tract infection involving nose and sinuses & throat.
Your tickly feeling in throat is due to 
post nasal drip, a characterstic feature of 
sinusitis. For confirmation of diagnosis and treatment point of view you have to do these investigations-
1- C.T.Scan of 
paranasal sinuses.
2- Culture and sensitivity of throat swab.
3- Gram's staining of throat swab.
4- Blood- CBC, ESR, blood sugar (fasting & post parendial).
Beside this a thorough examination of your nose, paranasal sinuses and throat along with clinical co-relation of inferences of above investigations with clinical findings one may reach to final diagnosis that may be accordingly manage.
Regards.