If using Kanger type single bottom coils, you'll need to wiggle off the chimney from the coil before dry burning. Otherwise you won't be able to see the coil.
Remove the loose flavor wicks as well. (I find they're usually good for trash at that point, and replace them with 100% cotton yarn. Others use other organic fiber, or just use 2 1mm strands of silica wick.)
And you may want to re-rinse the coil after dry burning and before putting it back together.
About drying... Yes, base and tube need to be dry, but if you're in a hurry, you can just blot dry the coil / head and go for it. The first few draws will be weak in flavor as you boil off the rest of the water in the wick, but it won't hurt it.
Or, just have a supply of extra coils, and pop in a fresh one while the old one dries. You DO need to have spares!
