Use your judgement on the stems. I cut the real thick stems. For me, the thick stems sometimes come loose and pull out. Then the smaller stems underneath seem to work there way out from the loose thread.  If you want fluff, cut the stems. I also use a head cement or superglue to solidify the tie.

Another thing that might help with some fluff, is to use some shrink tubing over your Coil wire, or simply take several wraps with the thread to create some cushion.