Thinking about how I write/portray Madgod Korone and I think the most important thing is that she's a physical manifestation of letting the intrusive thoughts win. She's not "insane" in the way that the original Sheogorath was portrayed, with implications of limited reality contact (which doesn't make sense to me in the context of a god of madness anyway) and a complete lack of logic (which makes perfect sense for how he came into being in the first place), but more like someone whose thought process has no "low" setting. She has no limiter and everything she thinks is automatically parsed as valid and accurate and correct because she's a god so she gets to decide what that means.
In the interim, when she's still settling into it, she can tell that this is "wrong," that she should be able to put a filter on her words and actions, but over time she loses that capacity, and eventually stops even contemplating it in the first place. Anything that any part of her wants to do, she's going to do, unless some other part of her (or Haskill) can come up with a reason very quickly as to why she shouldn't.
Her existence as Sheogorath is a big adventure in affecting the world around her in extremely noticeable ways, born from the fact that her choices and intentions wound up meaning basically nothing in the Oblivion Crisis. She didn't let herself move fast enough to stop Martin from sacrificing himself, even though there were parts of her that wanted to tackle him to the ground the moment he started his goodbye. Past that, people were going to remember her how they wanted to remember her, not how she actually was. They were going to apply their own assumptions to her motivations no matter what. They were never going to see her as herself, so why did she ever bother keeping so much of that self locked up where nobody could see it?
Korone got to Kvatch and jumped headlong into an Oblivion Gate with basically no hesitation, because the worst that would happen is that she died and nobody else was doing anything about it. After she lost everyone in Kvatch, she joined the Fighters' Guild to learn how to better operate with a group, in the hopes of that never happening again, and wound up in a conspiracy with her superior officer that led her to chugging a bottle of Extract Of Tree God Tortured To Insanity and eventually massacring the people responsible for sourcing said Extract, one of which was a former colleague. She then took everything of value from their bodies and their headquarters, every piece of armor and jewelry, every book and bottle of wine, and sold it all to buy herself a house across the square from the Fighters' Guild headquarters—because the irony of it tasted too good to pass up.
She's very much not mentally stable, but maybe she never really was. The difference is that she kept a lid on it before. After taking the role of Sheogorath, she doesn't do that anymore, and as a result she's more herself than she's ever been.
That includes going, "What if I dug up the ghost of one of Martin's crazy predecessors and fixed his brain so he can pass on to a different plane and Martin can meet more of the family he never knew? It'll be a little gift from me to him! Wherever he is!" Unfortunately, it also includes things like, "You looked at me with what I think was distaste and I didn't like it, so I'm going to turn all your bones into cartilage. It will last for the next seventeen seconds, or until the screaming gets annoying, whichever comes first. It's fine because I'll feel better afterward."
