Hi, Katie! Welcome to the forums!
I am not a mushroom expert however I have experience, both personal and professional, with anxiety disorders. I am so sorry your lawn has become a source of concern for you and hope to help a little.
Why you may be getting some identification wonkinesss: For starters, it looks like you may be using the Seek app? It is sort of the “little sister app” to the app most of us use, iNaturalist. The Seek app uses AI to identify. This is great for many, many common things, however mushrooms and fungi can be difficult for AI, as explained here by someone who is a mushroom expert:
I think the “Toxic” notes you are seeing are meant to dissuade any would-be foragers. You seem unlikely to forage for wild mushrooms, so I think you can set that worry to the side. As others have noted, you can pick up mushrooms even barehanded just fine. (I remember as a child doing so to see the pretty colors underneath.) The gloves you are using are not at all a bad idea though if they make you feel more settled.
As far as mushrooms themselves go, as others have noted, they are part of healthy soil. I would encourage you to reframe them that way and treat your floors and shoes no differently than you would after walking on your regular lawn. That might look like using a good doormat before you come inside, taking off shoes worn outside just as you come inside to switch to house shoes, or whatever your usual habit. But I would encourage you not to add steps because of the presence of mushrooms in the yard.
Saludos from Yucatán, Mexico! :)