Harmony uptime is a bit low at 81.6%. You can get that up to 95%+ just by using swiftmend more often (you had only 6 casts)
Nature's Vigil. You had only 2 uses, when you could've had at least 5 uses. Remember, even if you don't need the extra healing (like at the pull) - its also free damage to push the phase that much sooner.
If you have the leech trinket, you may want to swap out SoC for the leech trinket instead.
As for healing tips, make sure there's communication on who is using CD's for breaking shackles so you aren't needlessly wasting any. Make sure you are using roar during some of the later allure flames in phase 2 (and with multiple druids, you should each be taking one) as that will help cut down on doomfire damage.
Really on heroic its all about just having people execute mechanics properly. (like not breaking shackles just before demonic feedback etc)
Beyond that...rest of the problems really seem to be with your raid (people dying to doomfire - and too many low dps) - Also if you have a warcraftlogs instead of AMR, those can be more useful for seeing whats going on - imo.
Got logs from Warcraft Logs? I'm unfamiliar with AMR's logs and they seem difficult to follow. Some basic tips though:
-Aim to keep Harmony uptime at 100%, easily manageable by using Swiftmend often.
-Make sure you're getting maximum uses out of Tranq. If you're assigned to Tranq 1st Shackle, it should be up for Rain of Chaos in P3. If you have a different assignment, work around that to make sure you get 2 uses.
-Make sure you're getting maximum uses out of ToL. This likely isn't "assigned" a CD spot so use it where you find the raid taking damage early (Desecration Pillar/Shackles) but ensure you'll be able to use it a 2nd time in P3.
-Make sure to keep Mushroom down wherever the most people are stacked
-Keep LB up on both targets (if you have 4pc) 100%
-Trinkets: UFE & DP are best for progression. UFE Leech healing doesn't show up on logs but it is much stronger than SoC.
-Ironbark the tank when Doomguards are out.
-Channeled Mana Potion as soon as you transition to P3 isn't a bad time.
-WG less to save mana and instead use RJ & Germination. However still utilize WG during times like Shackles/Rain of Chaos when multiple people will be low often.