I have a mild cough that results in small amounts of hard yellow/ mucus coming up. I can barely talk but other wise feel fine. Should I see the doc?

Yes, if > 5 days. A viral cough, if it is going to "go away on its own, " should not last more than 5-7 days, and should not be "productive" (coughing up mucus). Hard yellow mucus means the mucus is dried out, either from decongestant medicines or if you are not drinking enough liquids. Even if you don't need an antibiotic, a productive cough and trouble talking could be from bronchospasm and need an inhaler.
Probably... If this cough has been present for 3 weeks or more, you should see your doctor. Possibilities include bronchospasm from any etiology, post-nasal drip, gerd or a combination of factors. Treatment is for each of these etiologies is different so it is necessary for your doctor to make the diagnosis first so the correct treatment is rendered.