Is there a way to make this edge look neater without using loop cuts? I am trying to sharpen these edges without getting the overhang artifact you see in the image. I was using the bevel modifier that works for some of the edges as seen in the picture. A loop cut helps this particular edge, but it also sharpens the outside wall of the model, which I don’t want - and my topology isn’t built well at all for edge loops (I had to do a lot of things to avoid tris and n-gons)

If I need to retopologize, I will. I would use tris if I needed to and doesn’t hinder the result but n-gons are not an option. If it’s helpful to know, I am going to bake this hi-poly res model to a low poly version.
For those who may be having the same question, I figured it out. When I removed the bevel on the outer edges, but kept the bevel on the other edges, the artifacts and problems disappeared. I was able to harden the outer edge by adding loop cuts to the walls of the inside extrusions.