Now this is not advise but kind of sharing of experience from a priest new to healing. So, I could be wrong so don't chop my head off. I can speak even if I am not an expert ok?
I don't do holy but I do disc and I do find it a struggle to keep my mana up. Here's what I feel should help though:
1. Understand that disc is mostly reactive healing in terms of mechanics and holy is reactive healing mostly in terms of target health. For instance if tank is facing and incoming high damage then something like spirit shell and greater heal will do him good. You do that too early that it goes off before it is completely consumed and you are wasting it. You do it too late that it is applied after he takes most of the damage then it will just go wasted and not get consumed completely. So, when I say reactive healing in terms of mechanics, I mean know the fight and the incoming damage to your target(s) and make preparations for it. If not then you will find yourself spamming flash heal to keep up your target and go OOM.
2. The reason why they are asking you to go crit is because disc is an absorption healer. The higher your crit the more powerful your shields the less you have to heal. The less you have to heal the more mana you save.
3. Half the skills is in managing mana yes. Knowing your CDs and making good use of it. It will take time and practice to learn to synchronize them to your fight and play style and tank's playstyle. Some of the CDs are mindbender, rapture, trinkets, hymn of hope.
4. Make sure your throughput is maximum with minimum number of casts. The less number of times you have to press your buttons the more you have mastered disc healing. For instance, I keep grace up on the target at all times. Or try to. I know you are only concerned about the mana but mana saved is mana gained.
5. Here are some of the things I do. Although I am still learning to keep my mana up all through the fight:
a. use renew. I know some may disagree but I use it and you can try too with the glyph. I feel the HoT along with the proc of divine aegis helps.
b. accept that you can't keep the whole raid up. You are not an AoE healer. At least not as good an AoE healer as Holy. It's the raid healers job. Your's is to keep the tank up and the raid healer is to help only if you need it.
c. Keep Rapture and mindbender on CD.
d. I spam holy fire every CD. I don't care what's the best way to do it because there's already much to handle and so I keep it simple. I just get my archangel whenever it is ready. Even if it goes waste. Overall it works in my favor. May someday i will work on improving on it but so far it's been good for me to take it as soon as it's ready.
e. A few seconds into the fight I pop spirit shell, inner focus and shield the tank good with greater heals and pop mindbender on boss. This brings me to full mana again and comapared to not doing it, I am at the same state except the tank is shield for good 15 seconds for any inc damage. I don't care if it gets waster cus I am still at full mana and mindbender will be ready next time I need it.
f. Now, I keep hitting heals ever 2-3 seconds to keep grace up and PWS to keep rapture up. Keep renew and prayer of mending up.
g. When big hits come, I make sure I have him shielded again and use infusion if spirit is not ready. After every big hit when you have healed everyone back up, there is small gap where no heals is necessary. This is where I pop hymn of hope. yes, early into the fight because that way there's a chance to use it again in the same fight. I am still at almost full mana.
h. I don't touch flash heal unless it is to save the tank from dying. And that happens if I screwed up. I think you should never need to use it if everything went right. I don't heal other raid members except with PWS or penance if they are low on health. I don't spam prayer of healing to keep the DPS from dying. If they die it's not fault. I can only help with renew at most of like I said penance or PWS. if i started healing the DPS I WILL go oom. disc healing builds up on a target and shields them after applying a bunch of things on them. I can't do that on 20 or 25 ppl.
Again, I said I am still learning and reading and practicing so take it as a fellow noob and I hope you get some good advise from an expert too.