Started coughing a week ago. Had other symptoms of a cold. Evrrything has gone away, but the cough. Should I go to the doctor?

Cough. The cough is usually the last symptom to resolve with a cold. I'd give it more time, unless your having any difficulty breathing, fevers or worsening symptoms.
Yes... Sometimes, after an infection resolves, the airways remain inflamed afterwards and bronchospasm can result. You may need a bronchodilator inhaler and perhaps a steroid inhaler to control the inflammation and bronchospasm until the inflammation resolves. Your doctor can evaluate you and render the proper treatment.