All drop away rests are going to make some kind of sound. The launcher arms have to stop against something and that something is going to make some kid of noise. No matter how much dampening material you use you will always be able to hear some kind of sound with a drop away rest. Or any rest that moves for that matter. Even TM hunter rests will make clicking noises when the arrow pushes down on the prongs and the rests snaps back up.

Adjust your rest so that your nocking point and rest are in the proper place and use some sticky foam, mole skin, felt, Simms pad, or waht ever to make it quieter.