If the generic works for you, that is great. But the dextromethorphan is a cough suppressant, which is the opposite of what is needed to remove persistent mucus - here are the cautions at Mayo:
The presence of other medical problems may affect the use of this medicine. Make sure you tell your doctor if you have any other medical problems, especially:

Asthma—Since dextromethorphan decreases coughing, it makes it difficult to get rid of the mucus that collects in the lungs and airways during asthma
Diabetes (sugar diabetes)—Some products contain sugar and may affect control of blood glucose monitoring
Liver disease—Dextromethorphan may build up in the body and cause unwanted effects
Chronic bronchitis or
Emphysema or
Mucus or phlegm with cough—Since dextromethorphan decreases coughing, it makes it difficult to get rid of the mucus that may collect in the lungs and airways with some diseases
Slowed breathing—Dextromethorphan may slow the rate of breathing even further

If ever you need to check the information about a medication or supplement, including proper usage, cautions, side effects and potential interactions, you can go to this link and start a search: https://www.mayoclinic.org/drugs-supplements
Sue