Go to the store and pick up some Extra Strength Mucinex. I take two every day and it really helps. Asthma creates a lot of extra mucus which can sometimes be very hard to get out of your system. You get the dry coughs that aren't productive.
The mucinex will loosen all of that up and make coughs very productive. I have very bad asthma and taking mucinex has made a huge difference, especially in the winter climate.
You don't need a prescription by the way.