I completely see why you love the name. Evanescence has such a beautiful sound (kind of reminds me of Evangeline) and Nessie would be so cute.
It’s also very tied to the band. I’m an Evanescence fan, Amy Lee’s voice is incredible, but it may be too strong an association and make it difficult for little Nessie to find her own identity. I also think the meaning “vanishing, fading away” might feel a bit too bittersweet.
Only you know whether that would bother you enough to dissuade you from using the name. At the end of the day, I can see how much you love the name and it has a great nickname and is established as a word in it’s own right. I don’t think it’s unusable, just very associated with one specific image.
If I were going to have a little Evanescence “Nessie”, I’d probably name her siblings:
Polaris “Posie”
Zosime “Zora”
Idlewild “Isla”
Octavian “Otto” or “Tav”
Bensiabel “Ben”
Castalian “Kit”