Have a runny nose and when I blow its thick mucous with coughing up a lot of thick yellow mucous for a week now. Should I get checked?

Yes if no better. Most uncomplicated viral colds run their course in 7- 10 days. If thick yellow nasal mucus is not improving after about 10-14 days, the possibility that a sinus infection has occurred is significantly higher. Nasal saline rinse and decongestants may relieve symptoms, but if not improving over next several days, antibiotics would be reasonable.
Yes. When a viral infection, which is what you probably started with a week ago continues into the 7 to 10 day period, it is quite likely that it has now become bacterial. Usually a viral illness will be clearly getting better by then. You may in fact benefit from an antibiotic at this time.