My MIL is a very finicky person and she likes to keep things in order in her own home. But as soon as she comes over to our place, no matter how clean it is, she manages to find fault! How do I tell her, she needs to STOP, without sounding rude or crazy!

That is typical of MILs, so do not tax yourself thinking too much about her. My MIL keeps her house in such a mess and when she comes over to our place, I wonder in what capacity she speaks about this thing dirty or that thing dirty at my place. I think, listen from one year and let it pass through the other. Do not let her nick-picking affect your mental peace. You can talk it over to your husband about how funny she behaves and feel a little light at heart. They will anyway go, so let it pass.