You suffered considerable irritation to the vocal apparatus from all of the coughing. Things will heal on their own with time and it's difficult to predict how long it will take given the severity of the damage that was done. Mild vocalizing should be OK, but don't try to force anything too much. If you are still having trouble after 2-3 more weeks, go and see a good ENT to have your throat looked at.
Also, a lot of people don't realize that a persistant cough is a sign of Asthma. I always had the classic wheezing and shortness of breath but my daughter has coughing spasms. Make sure that you have the appropriate medications to control the symptoms so that the coughing spells are controlled. Asthma is a controllable medical condition these days and should not interfere with singing. See a good allergist to get more information on the available treatments.
Alan Goldhammer, PhD Associate Vice President Pharmaceutical Research and Manufacturers of America