According to Dr. Google, thickened mucus is a side effect of aromatase inhibitors. Also, according to Dr. G, thick mucus is a primary cause of coughing. It's also a side effect of another anticancer medication called Kisqali. I take both. I have thickened mucus also, but no cough. My Med Onco says it's OK to take Mucinex (or generic) to keep the mucus thinner. I take the ER type--1 tab in morning and it lasts all day. I don't seem to need it at night, so I don't take it then.
You can also get Mucinex with a cough suppressant, but I'd try it without cough suppressant first. If your mucus is thinner, that alone may stop the coughing.
If Mucinex thins the mucus, but you still keep coughing, that might mean the cough has another cause. If that happens, please see a doc soon.
I hope this helps. PS regarding nurses: If you trust yours, fine. But I insist on seeing a real MD. Sometimes nurses are great diagnosticians but sometimes they minimize symptoms.