Is it normal for a toddler to still have drainage in throat that has to be coughed/cleared regularly a week after all other cold symptoms are gone?

It depends: Cold symptoms can last for 2 weeks, but are usually clearing by the end of the 2 weeks. If you're not seeing improvement in the congestion or drainage, your son should be evaluated by his doctor.

Postnasal drip: If it is due to a viral cold the drainage should resolve on its own. If the infant still has symptoms it could be that he/she may need antibiotics for purulent rhinitis (bacterial rhinitis) or it could be allergies if the postnasal drip/clearing of the throat continues. The pediatrician may recommend an antihistamine such as Benadryl (diphenhydramine) or a nasal spray if the child is old enough.