I'm 99% sure its RMA time for these phones. This is most likely a problem with the Hookswitch. After deploying Cisco IP Phones for the last few years, we (and many others) have found this to be the most common hardware problem.
There is a "Best Practice" for cleaning the Hookswitch contacts (link below) that does work on occasion. Usually the phone has to be replaced/RMA'd.
Best?? Practices for Cleaning and Troubleshooting Cisco IP Phone Hookswitch
There was also a problem with the 7.2 phone firmware that caused similar looking symptoms for the 3rd generation phones like the 7970/7961/7941. This could be rectified in the short term by booting the phone. If your issue can be fixed by resetting the phone and you are running 7.2 firmware, let me know and I will find you the Bug.
These are the paths to get to each CCX logs through CLI. They may be helpful if you are having issues accessing RTMT or downloading logs through it.
If you want to download them you have to prefix "file get " and you can add one of the options (re...