A Shadow Priest blog

Mana
I am going to assume you bloodlust in phase 3 and your goal is to use your Shadowfiend right before a bloodlust as this basically means a full mana bar.
Generally, what I do is use Shadowfiend at the third or fourth to last add in phase 1 so that I can enter phase 2 with nearly full mana. If you use Shadowfiend immediately when it's off cooldown in phase 2 you should get it back well in time before bloodlust in late phase 3. You might need to adjust this based on how quick your guild clears phase 2.

You want to Disperse sometime during phase 2 either when grabbed by a tentacle, feared into a crusher, brain linked with someone far away or simply when running without any tentacles in reach. There's usually no reason to visit a Sanity well unless you screwed up and got too low.
Use a Mana potion in Phase 3 if you ended up low on mana. Otherwise you should use a Wild Magic potion in early Phase 3 and keep refreshing Shadow word: Pain on Yogg-Saron with the added 200 crit rating from the potion for the rest of phase 3.


Tentacles
Ideally you wanna put dots on as many tentacles as possible, but there are some limitations. It's not humanly possible to keep a sane rotation and still refresh Shadow Word: Pain with Mind Flay on more than 3 tentacles, so don't bother with that. Re-applying Shadow word: Pain on targets you cannot refresh with Mind Flay is a huge mana drain and not any more damage than simply focusing on three tentacles. Always try to position yourself so you're in range of as many tentacles as possible. Abuse instant cast spells to reposition, be it Shadow word: Death or Vampiric Embrace. Never run from A to B without doing anything.
Which dots you put on which tentacles greatly depends on how many others are focusing it and it's health. Vampiric Touch on a Constrictor tentacle is a waste since it's unlikely to damage tick more than 2-3 times. Only apply Shadow word: Pain if you're able to target it fast and help damage it down or just ignore if it's already being zerged down.
Raid damage is fairly low, so don't bother with applying Vampiric Embrace on tentacles unless you're about to reposition and can waste an instant. You want to put your non spammable spells (Devouring Plague/Mind Blast/Shadow Word: Death) on the Crusher since this is likely the tentacle with the most caster debuffs.

Phase 3
Position yourself between Yogg-Saron and where the adds are tanked and face away from Yogg-Saron. Don't bother keeping track of Yogg's gaze bar - instead, time it by doing Mind Flay and Mind Blast at Yogg not long after a gaze to refresh Shadow Word: Pain and rinse repeat after the next gaze to keep it refreshed. With this positioning you dont have to keep track of a gaze bar and you will not lose any sanity. You wanna have Mind Blast off cooldown when refreshing Shadow Word: Pain since there will be more caster debuffs at Yogg and a critting Mind Blast means more Improved Spirit Tap uptime.
It's important that you dot up newly spawned adds as quick since you want your dots to tick as much as possible before it dies. It's rarely worth it to put a second Vampiric Touch on an add. You can do this phase with 1 Sanity.