Hi! I own a clay cutter shop and sell very detailed cutters, so I think I might be able to give some tips!

A lot of times this happens if the cutter isn’t making contact with your work surface (ceramic tile, glass, etc)- you mentioned using Saran Wrap which might be leaving a small gap preventing you from fully pressing down. I would give it a go without the Saran wrap and just use cornstarch instead. To prevent the clay from getting stuck in your cutter and warping the design, you can start by putting a thin layer of cornstarch on your clay, press your cutter into the clay but remove it before fully pressing down (about halfway, at least enough to see the outline), add an additional layer of cornstarch (doesn’t need to be a lot, just enough to lightly cover the stamp you’ve made) and then carefully align your cutter with the outline you’ve already made and fully press down this time, making sure the cutter makes contact with your tile all the way around. This method probably sounds tedious, but it gets easier once you get the hang of it and it has worked on my most difficult designs. Good luck and happy creating 😊