Lingering cough

So my DD was sick with a cold and I and her dad got sick as well. I however recovered sooner than both of them but I now have had a lingering cough for about a week. It’s not a dry cough, it’s a consistent need to clear my throat (break phlegm) and cough when I cough it’s a little cough which feels like I’m not breaking any mucus but very annoyingly consistent especially at night. I have started taking Mucinex , the morning after I started I was bringing up a lot of phlegm, however I still can’t stop feeling the urge to clear my throat/cough. Sigh should I just go to a doctor now for some antibiotics?