Holy and mana regen.

[orlfman]

So today I was running Zul'Aman and we made it to the last boss after 7 wipes and 4 different tanks. Each tank I had was taking on average of ~12,000 - ~20,000 damage from each trash mob. Almost every trash pull I had to sit and drink for mana (which I can say was annoying to every single tank I had). We made it to Daakara and wiped two times. The first wipe was because people forgot to kill the small cats and stood in fire from the dragon hawk. The second time we wiped because I ran out of mana in a minute and a half during the boss fight because everyone was getting near death from the cat and fire. After the second wipe on Daakara they quietly kicked me. I honestly don't know what I could've done. They were taking so much damage and the only way to get them all back up in time was to pop all my cool downs and cast nothing but flash heal, greater heal, and my AOE's like circle of healing which drained my mana like nothing.

My stats, unbuffed:
Spell Power: 6,437
Spirit: 2,947
Intellect: 4,186
Mana Pool: 84,762
Mana Regen: 4,218
Combat Mana Regen: 3,580
Haste - 12.58%
Mastery - 13.17%

All my gear has spirit on it. I have an average item level of 353 and 3 pieces of my T11, valor wrist, valor wand, justice boots, 353 cloak, 346 main and offhand and such. All heroic/359/378 gear. This isn't the first time where I've had a dungeon run like this, actually most are like this.. I run out of mana so fast and I really don't know what to do.

[Lackluster]

You said it yourself, the party forced you to use your expensive tools way too often, running you out of mana. Any healer (class, spec or gear) will run out of mana if forced to use their flash heals constantly.
The only thing you can do is try to anticipate damage, allowing you to use your cheaper heals for longer. Don't let the party drop low if you can avoid it, as that will force you to use your expensive heals.
Also, use your regen cooldowns as early as you can (without wasting any of it) so you'll be able to use them later in the fight again, don't wait till you're dry before using hymn and fiend.

Other than that, yell at people to move out of the fire and if they consistently stand in it stop wasting your mana on them.

[zakaluka]

Ask your tank nicely to use at least a little CC

If they refuse, who cares if people drop party. Do everything you can, and try to learn something from it. There are a couple tricks you can learn to escape triage mode even when the troughput demand is way too high.

For most of us gheal gets >25k hps. If you heal proactively (instead of reactively) you can chain gheals much longer where you'd normally be spamming flash.

[Priesthoodz]

My tip 1) Be sure to use your Holy Word: Serenity as often as possible.
Instant cast + a Nifty heal at only like 1.5k mana = awesome. Holy Word : Sanctuary in 5 mans is next to useless (imo) and the only positive that MAYBE comes from it is the 8 second cd on Circle of Healing.

My tip 2) Have you tried putting down a lightwell ? Soon as you put it down, say in party chat that people should be clicking it whenever they get the chance to (or get an addon/make a macro to do that for you every time you put lightwell down). Also, you'll want to put it down a bit before the fight so you get a few seconds to regen/drink for the mana you just used to place it. If you know your group is Pro about clicking Lightwell, put it down ASAP so you get to put it down again mid-fight and get another 10 Charges of it, or 15 if you're glyphed.

My tip 3) The way I used to play when cata came out and mana was an issue (for me at least):

Prayer of Mending, Holy Word, Heal, Circle of Healing.
Use Flash Heal/Binding heal only when you REALLY need it and if you end up casting 2 in a row, your 3rd (if you REALLY need it) cast should be a Greater Heal or Prayer of Healing (w/ Serendipity you'll get the quicker cast plus the cheaper mana cost)

This play style was difficult for me to adapt to after being well geared in Wotlk and then having trouble in Cata Heroics but with the gear you are at now, you shouldn't be having the same problems. If its a problem with the group standing in crap, then they need to get their own problems fixed before you can pinpoint yours.

My tip 4) You could also time your Guardian Spirit to save yourself mana.

If the tank is taking a lot of damage and you know you'll go oom trying to flash heal through it, Put up guardian spirit and start casting heal. The guardian spirit *should* proc bringing tank back up to 50% and your heal should be getting off which will add some more health. You've instantly saved yourself maybe 10-15k+ mana. The downside to this is that if theres another situation where the tank is going down quickly or is about to die and need GS for the emergency, you have no choice but to try and heal through it.

My tip 5) Timing Shadowfiend appropriately.
Someone already covered this, it is very important when having mana issues to time your Shadowfiend smartly.

My last tip) Re-Spec Disc.
Maybe you don't want to (I know I didn't), but I found it was easier to do 5 man dungeons as discipline. You'll have the added Rapture mechanic to help with your regen and Power Word:Shield can give you time to cast some slower spells or be good in emergencies.

Hope at least some of this can help you out.

[ThePenguinFreak]

My last tip) Re-Spec Disc.
Maybe you don't want to (I know I didn't), but I found it was easier to do 5 man dungeons as discipline. You'll have the added Rapture mechanic to help with your regen and Power Word:Shield can give you time to cast some slower spells or be good in emergencies.

I found it much easier to heal the troll heroics as disc with a lower gear level. Holy is a little bit lacking in tank healing until you really get a fairly high level of gear.

That being said, from what you say your gear is at, there should be no reason to be having such problems.
Like everyone above said, think about completely reworking how you heal people--I hardly ever use flash heal when healing 5 mans-- and know when people are going to need more healing than your basic heal spam.
However, Chakra: Serenity is almost always the way to go, EXCEPT in 2 fights in the Trolls: bethekk, and phase 2 daakara.
Sanctuary Chakra seriously helps heal through the heavy raid damage in either eagle or dragon hawk.
So basically, try to be more judicious with your expensive heals and rethink how you heal.