At last! I think I have the solution for all of us:
I've been in the same situations, except my iPhone 5S gone crazy even if not plugged...
I've searched a lot on the internet to find a solution, that's why I'm here:
I tried a lot of things and multiple restaurations: Even during the installation, touch screen made those weird stuffs...
I finally downloaded a brand new firmware for my iPhone 5S (iOS 7.0.4.ipsw) to restore it on my iPhone. It exists for iPads as well.
You can find it through a search by your own on the internet, it's easy to find 😉
It seems our problems are not an hardware ones finally, but seems to be due to some corrupted files at the root of the phone's (or iPad) software.
After downloaded, open iTunes after have plugged your device and press ALT+"Restore" (MAC) or CTRL+"Restore" (PC) to choose the firmware downloaded earlier.
wait for the restauration done... And voilà 🙂
Worked fine for me: No touch screen issues anymore.
Hope that helped.