You are pretty much right on target as to what you believe is your limiting factors.
You need more avoidance to be productive in a T6 setting. This can be fixed to some degree with 2.4 badge rewards. Its all fine and good to push resilience into your gear to push off the defense "cap". However, you lose a huge chunk of avoidance because of this.
Things you can look to upgrade easily and in no particular order:
Gloves: Like it or not, but the [item]Warbringer Handguards[/item] are probably going to be an avoidance upgrade for you. I know you already have the Bonefist, but your avoidance is so low as it is, any upgrade may be worth it.
Chest: [item]Chestplate of Stoicism[/item] is the best bet for you, although it does have a high price to pay at 100 badges. The other option is [item]Chestguard of the Warlord[/item] from Zul'jin. Both are huge upgrades in avoidance for you.
Shoulders: Your best bet here is to get into a Gruul's Lair run and pick up some [item]Warbringer Shoulderguards[/item]. Its the biggest upgrade for your avoidance. Your S2 shoulders are alright, but the upgrade from T4 is HUGe. Well... to be honest, any pure tanking shoulders are going to give you an increase in avoidance. While picking up shoulders from Heroic Sethekk or Kara may seem to be a downgrade in EH, the avoidance you pick up will serve you a whole lot better. You can also pick up the ZA shoulders out of the first box.
Rings: Replacing your Shade of Aran ring with the new 2.4 badge ring will do the most to improve your avoidance in this slot. You will not come out an incredibly far way ahead but the dodge you get will outweigh any defense loss.
There are other places where the new badge loot would help you. Yet, the 4 slots I listed are the ones I would myself concentrate on. Other suggestions might be... Replace the gem in your gun with a green def/stam gem. This may be a controversial choice, but it is a completely valid one. The amount of stam lost for the def/avoidance gain is a decent trade off in the slot. Do you have Moroes? Twin stam trinkets are really NEVER a good idea in my opinion. Get a defense/avoidance trinket and replace your Netherwing trinket. The only other thing I may consider is picking up a mitigation/avoidance weapon over the sword. I am not saying go get a Suneater, but KD is still a very valid choice. Your current weapon is extremely nice from a TPS point of view but from a pure mitigation point of view, KD would be superior choice. I do not think that you would even need to consider changing weapons with the gear choices above though. Until then however, you may think about it.
All in all, work on the chest piece from badges, then the ring. All the while doing your damnedest to get a pair of actual tanking shoulders. Running Kara and ZA is your best bet in improving your gear. Are you geared for T5 instances? As an OT you definitely are, as a MT you would more than likely get toasted on several encounters. With a little work you will be ready for Hyjal quickly though. So get to farming those badges.