Hmmm ... I also have a newer black stove, but don't have trouble with finger prints and smudges as much as I do with dust and "floaties" in the air. It sits directly in front of a window and when the sun is beating in, I swear you can see the air as it passes the black enamel! For this, I absolutely swear by my microfiber towels. They are incredible and hold like 2-3 cups of liquid (and yes, unfortunately we have had to test that claim).
I have wondered if it was safe to use that Dawn power stuff, since I have a pump of it under my sink. It totally removed the non-stick coating from a muffin pan I had, so I am scared to death of it, LOL. Andy, are you SURE it won't remove the paint? I don't think Jenn-Aire will believe me if I call and say my paint is flaking.
This Sheila Shine sounds interesting ... does it remove the gunk, or just shine after cleaning? I have gunk around burners I rarely even use!*
*Footnote - kids who cook!