It might have something to do with sores or growth in your vocal chords. I also suggest you go to the doctor, but not just for the cough. You should have them do a full examination of the problem, because it might be something serious.
Sounds like what my choir director has. I'd rest your voice a couple days and see what happens. However, the inability to take a deep breath without coughing concerns me. If you can't take a deep breath, you're most likely taking shallow breaths which can lead to something much more serious. I'd go to the doctor and ask for something to help with the cough.