Phase 3 is all about manipulating the raid to not die during Rain of Chaos. If no one dies during Rain of Chaos, you win. The banish, the adds, and the infernals all really don't matter because if you can't handle those very simple mechanics then I'm not really sure how you got to P3 anyways since they're all easier to deal with than P1 and P2. What you have to do then is make sure each mechanic isn't going to kill people during Rain of Chaos.
Wrought Chaos - if people are spread then this shouldn't be hitting more than one or two people each time it jumps. It's really not a big deal and you don't have to run around with it, just let it happen and anyone that it jumps to probably needs to use a cooldown if it's in the middle of Rain.
Shackles - There's only 2 ways to deal with shackles: either you break them before the rain or you hold them. If you break them before and have enough healing to deal with that, cool, but make sure your raid knows that so you don't have some bozo hanging onto it until right as the Rain starts. If you don't have enough healing to just break them and heal up, the people holding the Shackles will need to use some massive defensive CD (full immunities or 90% damage reductions) to survive both the DoT and Rain at the same time until the raid can get topped back off after Rain is over. If Shackles goes on like a Shaman or something that can't do much, they're probably just going to have to break it instantly before the Rain and you'll have to deal with it. The worst thing you can do is try to have someone that got Shackles before Rain even started attempt to live through Rain and then they die because when they die their shackle is broken anyways and that's just an instant wipe. TL;DR either break them before or have people with immunities or 90% reductions try to live through Rain while holding the Shackle.
Demonic Feedback/Rain of Chaos itself - This is just raid damage that happens and you have to deal with it. Cooldowns are kind of awful to use during this because only raid-wide things like Nature's Vigil, Vampiric Embrace, Ancestral Guidance, Revival, Tranq, Hymn, or Devotion Aura do much of anything (i.e. your SLTs, Smoke Bombs, Rallying Crys, and Barriers aren't really going to do anything). If you have no planned raid-wide CDs and the entire raid is out of the Nether, you're probably just going to die. Plan your CDs and just heal through it, make sure people use health potions and stuff.
Should healing be so bad that you can't even heal through Rain of Chaos normally, you can screw with the Banish groups in such a way that you end up pushing Archimonde to 25% right after the 2nd Banish goes off, which allows people inside the Nether to dodge the Rain damage and the timers will line up that way for the rest of the fight.
There's also apparently some ultra-cheese going around now where the 1st banish group stays in the Nether until the 2nd group goes and the 2nd group just uses the 1st group's portal to escape and you use the extra DPS to kill him before the 3rd Rain at all, but I haven't personally done that yet.