The best advice I can give you is to take the rest off the bow, walk to your kitchen, throw it in the garbage and get a good standard rest.

I had a TTSH on my Liberty and through hours upon hours upon hours of messing with it, I took it off and put on a N.A.P. QuickTune 3000. I had the same problem as you. I adjusted that rest in every single way possible and could never get the fletching contact to go away. I would get it to go away for a little while then it would be back.

If the justice has the same center shot as the other new for 04 patriots your horizontal adjustment will need to be all the way over if you do not have the extended horizontal bar.

The center shot is real close to 15/16" and the standard horizontal bar will not get you there. Cock up is almost a must with the TT and the I-cam, not sure about the "E" cam though.
One trick they were using at my pro-shop was cock-out. I couldn't see how that was working if cock up wasn't doing the trick. Might want to try it.

From my own experience and from what I've read you will need the string served into the down cable and adjusted to where it's keeping the arrow up every bit of 2.5" of the last part of the draw stroke. Maybe even more.

None of these are absolutes, just some suggestions if nothing else is working.

I lost patience with the TT on my patriot and was able to use a drop zone, which allowed for fletching clearance.

If your pro shop guy tied the cord around the cable slide then I would highly suggest you find a new pro shop guy.

The rests cord should be served onto the downward moving cable just a little below level with the rest (1 inch) in order to keep the cord from interfearing with your fletching.

DO NOT put the cord through the cable. This can cause premature wear of the cable where the cord runs through it.

You will probably have enough cord left to serve it correctly if it is tied to the cable slide.

I must disagree with big bulls, trophy taker specifically states to put the launcher cord through the downward moving cable. I have never noticed any cable wear. The launcher cord does not move or slide through the cable enough to cause noticable wear. If you want expert advice call trophy taker, explain your situation and they will coach you through it.

Okay, double checked this morning and the rest string is mounted in what appears to be a tab just for that purpose on the far side of the cable slide. The string is "capped" with a tiny string nock to keep it from pulling through the tab. Now I am really confused. Is it mounted wrong and I need to change that or will this rest just not work on this bow? Fortunately I still have my dependable GKF as a back up if I can't get this straightened out. I was disappointed in how noisy the rest was this morning too. It was in the mid 20's when I was shooting and I was getting a strong metallic sound from the rest. Anybody else have this problem in cold weather?

you need to put some felt or moleskin on your shelf to quiet the thud the launcher arm makes when it makes contact.

first thing to do is put the cord between your bus cable from right to left, if you're right handed, about 3/4 inch below where the cord attaches to rest.

pull the cord tight, where its about to pick up the launcher arm.

draw your bow, let down, pull about 1/8 of an inch more through the bus cable, thus removing 1/8 inch of slack in the cord. this is the most possible amount of rope you will need.

cut the cord

mushroom the end.

tie serving above the cord for about 1/2 inch so the cord can't slip up, it won't try to slip down.

you should be good to go.

check your centershot, horizontal and vertical. you should be 1/8 inch nock high and dead center horizontally. if you're correct horizontally and tie to down bus cable you won't have fletching contact from either of those shooting cock feather up.

you want your drop away rest up as long as possible. just missing fletching as it falls.

you can pull xtra cord through to keep your rest up longer and use a brass nock to secure the rope on the other side of the cable. chalk your vanes. if you get contact with the arm let a little rope back out until you have no contact.

that's just how i would do it.

You either take my advice or not. It really doesn't matter to me. It's your cables.

I didn't say that it would definetly cause premature wear but it certainly can as I have seen it on more than one bow. My own included with a TKO rest served in with the cord inside the cable.

Some cords are fairly coarse and when pulled tight then released the cord moves back and forth inside the cable and rubs the inside of the fibers slowly wearing at them.