Yeah you want to use cooldowns straight away on the pull. BBoY and PBI should generally proc when you double conflag. Sometimes PBI will take a couple of incinerates, but 9 times out of 10 it will proc on or before you fill that second ember.
For Destruction, Dark Soul gives 30% Crit which is kind of poor compared to what Affliction and Demonology gain via 30% Haste and Mastery, and so popping it early is advised primarily from the increased ember generation that extra crit provides. It's ok to have dark soul fall off and still have 2-3 seconds left of PBI where you can get off 1 last chaos bolt. Your chaos bolts gain more damage from a heroic PBI proc than they do from Dark Soul, for what its worth (not like Ascendance or Stampede which provide monstrous amounts of damage).
Later in the fight though I would wait for PBI to proc before using Dark Soul, generally this means delaying it a few seconds which is fine.
Similarly, the 4pc is basically a mini dark soul each time it procs, providing great ember generation, but also giving a nice 15% boost to Chaos Bolts damage. The way to use this is to hover around 3 embers, generally using the proc to bleed one chaos bolt + refresh immolate if appropriate.
Code:
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Above is the weakaura for the 4pc, 5 second buff with 10 second ICD. Make sure you're tracking it along with your ember bar, as it can allow you to do stuff like casting a chaos bolt at 29 ember bits to gamble that your immolate will push you over to 30 mid cast, proccing your 4pc and giving you a greater window to utilise the crit bonus.
Took a very quick look at Fallen Protectors (raid starting in a sec). Immolate uptime was 77.7% which should be higher. Roll immolates on the bosses plus any big adds from the Desperate Measures. Noticed some FnB Immolate uptime too so I'm guessing that may be the reason. You did great on IJ so seems that that isn't an issue.
I do think Rain of Fire could possibly have been used more too.
Cooldown usage was spot on, and Havoc seems to have been used once per minute so means you were glyphing it. I'd advise against the glyph here to be honest to give more chaos bolt/shadowburn cleave.
Also would say to take AD here over KJC. Any movement is only slight shuffles which fel flame can cover. Getting an extra dark soul when they're in execute range is pretty sweet.
Can't really spend any more time looking at it, but I would advise not to shadowburn snipe the desperation adds, as they dont die and so dont refund embers.