Sounds like you don't have enough flavor wick after the dry burn. That will often cause the leak to the battery that you describe. Pour the juice back into your bottle, wiggle the chimney off and check. I use a full piece of Peaches and Cream yarn with generally good results on my T3S, Aro and most other bottom coil toppers. (Some times, with some Aros, a full strand plus 1 or 2 individual strands is needed, the P&C is 4 strands) Lay it across the main wick, put the chimney back on, and trim both sides even with the base. then fluff them out so the gap is filled with wicking.
It might be, too, that when you had them apart to clean you disturbed the main wick. If that's the case, you'll need to replace that. You can use the P&C for that, too, although you won't be able to dry burn

Use any small pointed tool (toothpick?) to work the silica wick out, piece by piece. Be gentle, so you don't distort the wire of the coil. Once it's all out, thread the P&C through. You can wrap one end of it to a point with teflon plumber's tape, which comes off easily.
Or you can use any other cotton for wick. Many use cotton balls, teasing out enough to roll into a "thread." For the coil, the wick should have just a slight resistance to being pulled back and forth.
Too tight and you'll get dry hits. Too loose and it will gurgle and leak.