Removing the old wick might distort the coil...
I find that the wick itself usually does fine for quite a while, with periodic dry burning to de-gunk both it and the coil itself. Use small needle nose pliers to wiggle the air tube off. (I put the head back on a base for a better grip.) Remove and, usually, toss the flavor wicks, then dry burn. Rinse again.
I just use a strand of cotton yarn as replacement for the flavor wicks. You can use the same for the main wick, but of course can no longer dry burn. (Well, you can, and that will remove the cotton wick for you!)
But if you can get the wick out without deforming the coil, you can also use a floss threader, like those used for dental bridge work. They are extremely thin plastic needles with huge eyes, easy to thread in the new wick. 1 mm silica, or half a strand of cotton yarn, should work fine.
Or try wrapping the end of whatever wick material you want with teflon plumber's tape. It wraps tight, and comes off easily.