I have cleared A3S and I can give you tips for the full fight to complete Syzygian.
What he said is for the most part accurate. I just have a different CD management for P1 and 2 because I don't have Foresight cross-classed, I kept Mercy Stroke because it can help on P3, and Foresight is garbage anyway. What I do is :
P1 : I drop Grit exactly like on my first post, which is after my 3rd combo (I do Power Slash - Delirium - Power Slash), then I pop Shadow Wall for the incoming cleave, Blood Weapon and stuff. I pop Dark Mind + Shadowskin for the splashes + the last cleave.
P2 : I pop Living Dead for the very first cleave, then I totally Plunge the Wash Away and never had any problem doing that even with DRG jumping and NIN Shukuchi-ing. We have fixed placements for Digititis so us 3 don't interfere in any way. When Digititis finishes casting and the hand does the animation, I pop Awareness + Dark Dance, then Convo. I generally have Reprisal right before the first cleave, or right after. After that, Shadowskin becomes available again during Equal Concentration, with Dark Dance. I use them both, then Reprisal mostly for deeps here. I use Dark Arts + Dark Passenger on both hands right during Equal Concentration's cast (Dark Passenger becomes available again at this precise time), then it becomes available again right at the end of the Hands of Pain cast, so I do it again here. I totally use Salted Earth during Equal Concentration tho because "don't AoE" is a bad advice. You hit both targets anyway, so their HP difference doesn't change at all. AoEing the two hands when they are close to eachother is a very very very good thing to do because it helps pushing the boss's HP when both hands reunite. You want the boss at ~66% when he transforms into a tornado between P2 and P3. I put Grit back on right before the very last cleave, going into P3. I also drop Darkside as soon as the boss is untargettable.
P3 : I'm very very low on MP at the beginning of P3 because I DA SE the living shit out of the boss + DA DP twice during P2. I start with a Hard Slash on the boss before the water puddle spawns, this way I can directly Syphon Strike the first add. I put Darkside back on right before this Syphon Strike also. I use Blood Price because there is a lot of very tiny raid-wide damage + the little water bubbles hitting random people, and Blood Price slowly helps refilling my MP and wouldn't be used anyway, so it's basically free MP even if it's not a shitton. I seriously recommend doing that. I Shadow Wall the first Tether, then DA + Dark Mind the second tether, and drop Grit instantly when I take the tether's damage. You should get a second Blood Price before the second tether explodes if you used it at the very beginning of the phase. I use Blood Weapon after the Ferrofluid symbols have spawned on everybody's head so that way if I have it, I don't waste my Blood Weapon and only use it after Ferrofluid. I use DA Carve & Spit for the last add wave (the 4 big ones).
P4 : As soon as the boss becomes targettable again, the very FIRST thing I do is FUCKING DELIRIUM combo because the boss immediatly starts casting Cascade. Your team has a MNK if I can recall, so he should be form-shifting to Coeurl before the boss becomes targettable, to refresh GL3 ASAP then immediatly Dragon Kick since he'll be in Opo-Opo.
Blood Weapon is available again here, I use it asap during my Delirium combo : It will end very precisely during the Ferrofluid. I stay out of Grit for the first 3 stacks, I take the first cleave naked, then pop Shadowskin + convo right during the protean waves, this way I have them for the next two cleaves. For the last cleave, I put Grit back on during Digititis and I generally also have a Reprisal proc'ed from Dark Dance. Cascade immediatly follows, my WAR bro Provokes, so I immediatly drop Grit. Grit is used like an Inner Beast here. I also make sure to keep a Low Blow for the Living Limb; I stun him ASAP, DA Carve, DA Dark Passenger on both boss + Limb, Sole Survivor, do one DA SE, and he's dead. It's generally time to reapply Scourge on the boss when the hand dies. I use Blood Price right after Digititis, because there is 6 Splashes incoming, so it's nom-nom MP and helps because it's time to re-activate Grit in order to Provoke the boss back during Cascade which comes right after this. I have my tanking stance for the first cleave here, then drop it ASAP and pop BW and stuff. I take the 2nd cleave with Shadow Wall, and my WAR bro Provokes the boss back, this way we take one tether each and have only 2 stacks, which permits us to tank out of tanking stance more. I follow the move for Embolus; the second tether happens after the cleave. I have the time to dish out a bit of DPS, get that Delirium on the boss, put Grit on, then the tether spawns, I get it, DA + Dark Mind GG2EZ. You'll take a shitton of damage here tho, because you have a Sluice + Drainage exploding at the same time, so your SCH should adlo you, even with Grit + DA DM. If you don't have an Adlo, you might wanna add Shadowskin, but it will force you to tank with Grit later. I take the boss back during Cascade, and the phase restarts in the exact same way. I just don't have the exact same CDs available at the same time, but it always comes up nicely so I don't die. If necessary, I stay in Grit for an extra cleave or so, it depends. Anyway, you have DA DM for tethers; you can use everything else for the cleaves. You have an incredible freedom in terms of CD management as a DRK MT here. If needed, don't be afraid to use Living Dead a second time during phase 4.
So, here it is. I don't know if it's the most efficient way to tank A3S, but it's how I got used to it with my team, it works very nicely and we get some gud tank DPS numbers. I generally end P2 at ~1150-1200 DPS, and the whole fight at ~950. We got our clear pretty recently tho, we sure have a lot of improvements to make (and gear to get).
My gear is : Thordan EX Claymore, 210 Eso head, 210 Eso chestpiece, 210 Eso hands, 210 Eso belt, 200 Void Ark pants (need dem Gordian pants bruuuh), 210 Eso boots. For accessories, I use 2 i150 pentamelded STR accessories which are the neck and one ring with the STR Gordian earrings, the STR i210 Eso bracelets, and the STR Gordian ring.