The reason so many people think there is a problem is because they are so used to having infinite mana and corruptions that just carried them through everything in BFA. Claw and Lucid dream were insane and acted as a crutch for a lot of players.
Okay, so let’s go over the good and the bad from just this one Inerva Darkvein log (Huntsman damage pattern sucks for like every healer, so we will skip that one).
Right off the bat I can see that you are using Kleia instead of Pelagos. Kleia is indeed better than Pelagos but only when you have 34 renown and unlock her last ability. Until then, just use Pelagos and make sure you use Weapons of Order whenever you have a good opportunity to do so.
You are running Jade Bond and Grounding breath. Now, Jade Bond is completely fine but it is only worth running if it gets you an earlier(and better) Celestial window, or if you get an extra celestial cast out of it. For most consistency, try running resplendent mists or the revival conduit. Keep in mind that pretty much all of our conduits suck 
I use resplendent mists for everything (it’s pretty good in m+ actually).
As for Grounding Breath: This ability looks really good on paper, but it has a 1 minute PPM and a 1 minute ICD. This means that you can only get ONE proc per minute. Instead I would suggest the fortifying brew absorb shield one. It’s nothing special, but it is better than Grounding Breath
Enchants & Gems: You are using the + mana enchant on your chest, however this enchant is only worth using for shamans because they gain % mana on crits. Instead you should be using the +30 stat enchant as it is currently better than the alternatives for us. Main stat is insanely strong right now, so you should prioritize it when possible.
Cloak enchant is good, so just keep enchanting +20 stamina +30 leech on any new cloak you get because leech is overpowered.
You are using +10 enchants when +15 enchants exist. It’s minor, but everything helps.
Now for the gems. You should replace one of your crit gems with the +7 int gem, and probably the +6 int gem as well for your other socketed piece. Intellect is far better than secondaries right now, so this is definitely the play.
Eternal Grace is not bad, but Celestial Guidance is just way better. On my Mythic Hunger Destroyer kill I had a 51.38% uptime on it. That’s about 2.6% main stat on average and it affects your damage too (which is a big part about playing Mistweaver)
Before I get to your spell usage, I’ll go over your talents real quick.
All of them look fine, except for your first row.
Mist wrap is okay, but it is only better than Chi Burst when your Yu’lon accounts for like 20-25% of your healing. Chi Burst is insanely good, and it can easily do 4-5% of your healing if you use it on cooldown when there’s damage going out. 4-5% free healing from a talent is hard to pass up on, so make sure you use it.
And now we are getting to the interesting parts.
You are running Rising Mist which is absolutely the play for every fight. This talent is insanely mana efficient, and not using it properly is most likely the main reason why so many Mistweavers are complaining about a lack of mana. With Rising Mist, it is absolutely vital that you use Rising Sun kick as close to “on cooldown” as possible. This is even more important after you have used essence font or yu’lon, because it extends your hots and gives you a lot of healing for just 750 Mana (RSK mana cost). If you are not using RSK enough, then Rising Mist will not be as strong as it has potential to be. it’s not easy to get used to this playstyle change, and I myself am still using RSK far too little so just keep practicing. I would recommend getting a weakaura for this, as it really does help.
As for legendaries, your three best choices are:
Tear of Morning
Ancient Teachings of the Monastery
Invoker’s Delight.
Tear of Morning is your most reliable “all arounder” because it’s not bad on any fight (or amazing, except for on Stone Legion Generals). It’s not very noticeable, but is considered to be the best legendary for the majority of players.
Ancient Teachings of the Monastery (aTotM) is very good as well, but it loses a lot of value if you are not using your Tiger Palm, Blackout Kick, and Rising Sun Kick enough. You should be DPSing whenever you don’t actively have to heal. This legendary rewards doing something you should already be doing in the first place. On your Inerva kill it did 9% of your healing which is just crazy when you consider it’s free.
Invoker’s Delight is my personal favorite, but it’s not used very often and I would not recommend crafting it if you are not comfortable with Celestials. It’s insanely good on Sun King, Sludgefist, and Denathrius. On Heroic Denathrius I have reached peaks of 20k sustained HPS for 25 seconds with 40% overhealing while using this legendary.
You could use the breath of yu’lon legendary, but it’s just not as good as the other choices.
On Inerva you used 21/36 Renewing Mist casts, 7/11 Thunder Focus Tea casts, and 20/34 Rising Sun Kick casts. ReM and TFT should be used on cooldown, as they are both very mana efficient and will provide you with a lot of healing. Essence font into 2 Rising Sun Kicks is a ton of healing for a relatively small amount of mana. Rising Sun Kick is a bit harder to use on cooldown, but you should still try to cast it more than you did on this fight. Keeping on Renewing Mist charge ready for a big mana tea burst window is also a good strat.
Your mana tea’s could definitely use a bit of work as well. If you are not currently holding it for a celestial, your standard “rotation” is this:
Prepare by putting out Renewing Mists > Essence Font > Mana Tea > Spam vivify until the last GCD of Mana Tea > Essence Font again.
As for Yu’lon, this is what you want to do without Innervate:
Essence Font>Yu’lon>Mana Tea>Enveloping Mist until Mana tea is over, then use TFT RSK (1 RSK), 3 Enveloping Mists, and then RSK again before using Enveloping Mist until it’s over.
Keep in mind that you will never have to spam this many spells during Yu’lon as there’s just not enough damage in Heroic and most Mythic bosses to actually do so. It also costs a ton of mana, so unless you have an innervate I would not spam throughout the entire thing.
For Chi’Ji you want to cast enveloping mist after getting a single stack of whatever that buff is called. If you are tight on mana, you can get 2 or even 3 stacks to make it even cheaper. Chi’Ji is generally the play on most bosses, so I would advise you to stick with that.
A small thing to note is that if you use Tear of Morning and have 4 Renewing Mists out, Vivify is more mana efficient than Essence Font. Mana Tea also becomes a lot better if you have 2-3+ vivifies out when you press it.
You are using TFT on RSK which is good, but it’s fine to use it on Vivify as well if you are in a situation where mana is tight. Using it on Rising Sun Kick is generally the play though.
You understand that EF and RJW are your best spells, and I can tell that you are pressing them relatively often. Keep this up, and work on the other things and I guarantee you that your mana will look better.
Okay, the very last thing I’ll go over are stats.
If you are playing Rising Mist, then try getting some more haste. You can play it just fine with low haste, but it becomes a lot better with a decent amount of haste and is just more fun to play. I’m personally aiming for 19-21% haste before I start prioritizing crit again. If you want to play it safe though, then just stack crit and vers.
My fingers hurt a bit, so I apologize for any slow replies.