
Maybe because I haven't exactly been keeping up...but why exactly is she so obsessed with finding out who L is? Didn't she just want to live peacefully as a commoner, not that that's been fulfilled either.

Umm i think it may be because L is sort of disrupting her plans to just live peacefully without any attention, for example L is trying to prompt a war for some reason between the kingdoms i think? And in the process hes kind of dragging her into this by doing things that point to her. And she also never knew about this L character at all from the game so shes suspicious since he seems like such a powerful and significant character.

L is trying to drag her back to her original place since he prophesied the future. Remember that in the original story, Fel is the MC who saves the country but it was too troublesome for her to carry that burden so she wanted to become a commoner. She's afraid that L will try to drag her out of hiding with the war so she can become the heroine like she was in the original game
This is one of my favorite manga. Every once in a while, I come back to read it and I've always loved it.