I think the others might have overlooked something in your post that might be explaining a large part of your problem - battle effects.

Not all of us love all the flashy lights as they can be a distraction from what we're trying to do and when there are too many they can make it hard to see AoE telegraphs.

Go into Character Customization - Control Settings - Character and change the Battle Effects settings at the bottom of the window. Personally I use Show All for Own (Limited will work if you find it still too much), Show Limited for Party, and Show None for Others. If you've made multiple characters, you'll need to do these changes separately for each character.

It makes a huge difference in being able to see what's going on. Hopefully that will make dungeons a better experience for you.

As for the number of AoEs, just like WoW you'll get used to all the fight mechanics after you've done the dungeons a few times. Give yourself a chance to learn the content. Most of us (always the occasional jerk out there) are understanding when it comes to others trying to learn the content.

I know you mentioned that you're almost level 60 so you're no doubt into the dungeons that do not currently have Duty Support (SE is slowly getting it added to older dungeons) but you can always use it again to practice dungeons you felt had mechanics that were more difficult to learn.

Also pay attention to Espon's good advice about how enemy abilities snapshot. You generally want to move right away even if it's interrupting your cast until you learn the mechanic timing and will know if you have time to finish or if you still need to move immediately. There is one exception to the snapshotting I've found and that's Gaze mechanics (when there's a red eye over the boss). You generally want to make certain the animation for the spell has completed before turning back to look at the boss. Otherwise, just move out of the orange telegraph (if there is one) when it appears, stay out until it disappears, and then you can move back when it's gone regardless of the boss animation.