Man, those Aasimar look incredibly powerful. I love them, but I'd have to be very careful allowing one in game I think.
I actually think they look a bit meh, especially the Fallen Aasimar. One round of fear is not much; minor healing once per long rest is okay, but doesn't compare to the variant human with Healer feat. The bonus necrotic damage, eh... It would be more impressive as a bonus action, but as an action, it will take three or four rounds for you to even break even on damage compared to just using that action for attacks. The weaker your normal attacks are, the better Necrotic Shroud is; a 10th level Sorcerer using Fire Bolt might break even in only one or two rounds, but a feature that turns your cantrips once per day from weaksauce into something competitive with a fighter's weapons is nowhere close to broken. It isn't terrible, but it isn't transformative either the way Nimble Escape or a feat like Mobile, Sentinel, or Warcaster is. The Feral Tiefling's Wings from SCAG are far, far better than anything the Aasimar gets.
Either I'm misreading you or your misreading the extra damage. The extra damage is added once per turn on top of what you normally inflict. Aside from the action to enter it, it's free damage.
As written from that posted image I agree, and it doesn’t seem like there’s a lot of room for interpretation. As such, I’d ban this in a heartbeat, that firepower amp is insane. Even at third level it’s basically doubling your modifier to damage in most cases, and while it doesn’t synergize as well with multiple attacks, they do increase its reliability. The fact that it doesn’t require any effort to trigger and comes with free CC out of the gate is just gravy. The fact that it’s necrotic damage is a slight downside, but even with the plethora of undead in D&D this still seems too good.
If you assume that a player has even 50% uptime on the ability, at 10th level that’ll be an extra 50 damage that encounter. Are there any other racial abilities that can contribute that level of combat strength?