Cold symptoms or do you think something more?

Little one is 5 months old. He has a runny nose, is very sneezy, and a wet cough that happens mostly in the morning and the early day and is sporadic. When he does cough though, it sounds horrible. He does not have a fever. He is also teething hard. I called the doctor and she chalked it up to a common cold, but I’m so worried about his cough.