Did they give you albuterol or a rescue inhaler? I think that would help a lot. Also you might try otc mucinex. I wonder if the medicine they gave you isn't the right one for you.
I don't know if it will necessarily help you, but I get good results for the same symptoms by using a Neti Pot: http://www.himalayaninstitute.org/Netipot/NetiPotGateway.aspx
I know it's a little weird, but it really does help me get rid of that excess crud in my nose & throat. Good luck!
Throat clearing is usually a result of postnasal drip, not asthma. Inflammation of the nose and/or sinuses is responsible for postnasal drip. Routine use of a nasal steroid spray can be very effective in the treatment of postnasal drip. It decreases inflammation in the nose and sinuses along with mucus production. When a nasal steroid spray is used regularly it can prevent postnasal drip. However it does not provide immediate relief of symptoms. It may need to be used every day for several weeks to months to notice improvement in the symptoms. To get the most help from any nasal spray first do a nasal wash to remove mucus from the nose and sinuses. To learn more about this technique please read our nasal wash treatment information by copying and pasting this address
http://www.nationaljewish.org/disease-info/treatments/alt-ther/nasal-wash.aspx
into the address window of your internet browser. Share this information with your doctor to see if you would benefit from this daily treatment.
Another possibility is that you could be having gastroesophageal reflux disease (GERD). You should check with your doctor on that possibility.
Have you seen an allergist or pulmonary specialist? With asthma, you don't get mucus unless there is an infection going on. The asthma itself causes chest tightness due to the narrowing of the air passages. You may need additional medication or a change of the meds you do take. Don't try cough meds because there probably isn't any mucus there to cough up. You need to have a pulmonary function test done to see if you are moving enough air.