I know this doesn't address the anxiety aspect of it,....but,...
I'm just coming off having the flu for 3+ weeks and I had the most horrendous problems breathing and the coughing fits were so harsh, they were painful. I'd even wake up with them.
If your doctor says it's OK, you may want to look into a cough syrup with an expectorant in it. Coughing is important (it's the "not coughing" part that's not good). It's a healthy sign that your body is downright unhappy with this nasty thing happening to your lungs and it's trying very hard to kick it out.
Getting the phlegm broken up and expelled is very important. If you do not drink enough, the phlegm is too tough to come out easily and you'll cough harder, so as Larry said, lots and lots of water!
You'll kick this!
M.