To add:
With very few exceptions (mostly continuous damage over time attacks), the animation isn't dangerous. The moment the cast finishes, it is generally safe to move into that aoe. This works both ways, moving late, even if the animation didn't hit you, is still moving late. Understanding snapshots will help you immensely.
This game isn't about reacting, it's about spotting patterns in how mechanics are ordered and overlapped. The sooner you stop playing by reaction and start playing by prediction, the easier things will be for you.
It's okay to stop your rotation and focus on mechanics. Yes, even in savage and ultimates. Especially in savage and ultimates, actually. First live, then get the uptime during the mechanics once it feels easier to you.
Do "/petsize all small" to make summoner pet animations much smaller. Change your animation settings for party members to limited, and for everyone else to off. You'll be able to see the boss and actually react.
Actually pay attention to the boss, to things that get placed around the arena, and again - predict, don't react.
You don't need to get super sweaty and know a perfect rotation. Most jobs work on the principle of overlapping all your buffs with other people's buffs, using your highest hitting skills during your buffs, and using your skills on cooldown. Do this and damage will not be an issue until later savage fights.

You should know this by now. Either from other players (including content creators), or experimentation. If you didn't, you do now. Take it bit by bit and incorporate this stuff into your gameplay, and you will do better.