Welome to TMA!
There's a few good examples in The Peak District which isn't far. Gardom's Edge is probably the best example, although there is a bit of a story to it. I think the TMA posting probably explains it. It has never disappointed me seeing it.
There's also a crackin cup marked stone at Stanage, close to Eyam Moor sites.
Don't know of any in the direct Staffordshire/Cheshire area though. Where are you based? I'm in Stoke.
Megadread can probably tell you more as he's a great resource for all things Peak!
There's also my book (warning: self promotion alert) The Prehistoric Peak which you can find details of at http://www.andrewjohnstonedesign.co.uk , although the book is more a general guide to the prehistoric monuments of The Peak District rather than rock art specifically, but it does give clear descriptions on how to find the above sites, not that either are particularly hard to find.
(Sorry, couldn't resist shameless plug)
There's also some carvings at Rowtor Rocks at Birchover. Although I believe their authenticity/antiquity is questioned.