Hi, welcome to the forum, this feeling of hair in the back of throat sensation is known as Globus sensation. This is mainly seen in gastro esophageal reflex disease or hyperplastic tonsil.
You need a throat swab for culture and sensitivity also endoscopic swab of sinus to rule out sinusitis. You need an appropriate antibiotics and anti-inflammatory drugs if infection is present. You may also have to get 24hrs gastric PH evaluation, and upper GI endoscopy to rule out acid reflux. Usual treatment consists of a course of proton pump inhibitors against prescription.
The other possibilities are abnormal upper esophageal sphincter function, stress or psychological abnormality causes this sensation due to altered upper esophageal pressure.
Firstly I suggest you to consult ENT surgeon for possible hyperplastic tonsil and further management. Take care and regards.
Your cough can be from many things. It could be that the infection is not completely gone, allergies/PND, dry air, particulates in the air, acid reflux, a medicine you are on... It is really impossible to for us to say exactly.
It might be a good idea to see your doctor again to make sure the the infection is completely gone. You don't want that to come back as "super bugs" develop from incompletely treated bacterial infections. Certainly if the cough continues you need to see a dr.
Feel better.
If you recently started a medication for elevated blood pressure, that's a likely cause.
Bad tonsils with crypts can cause a tickle.
If it's just dryness, I'm sure you can get a throat spray that moisturizes or anesthetizes to calm your cough.