If you do have juice down the center shaft one way to clear it. Wrap a bit of paper towel around the airholes at the base loosly and blow hard short burst through the drip tip. The liquid should end up in the paper towel.
As for coil life. It varies a ton how long you can get from a coil. Very sweet or dark juices will gunk up a coil in no time causing it to need to be changed more often. I have a favorite that gunks things up so fast I have to change the coil every 60 mL. If you let the juice get too low in the tank you might have let the coil go dry and burnt the cotton in it (not in temperature control) and that is an instant replace the coil. How much you vape can also affect how often you change the coil.
Once a week is not that unusual. Some folks can get longer depending on what they are vaping and how much. 3 weeks would not be all that common from my experience.