Okay, I have this exact same problem since the last stable update with this exact same network card. I'm running Cinnamon DE.
In /etc/modprobe.d I have a blacklist for r8169, so that should not apply to my card/driver.
When I try to
sudo modprobe r8168 it says:
modprobe: ERROR: could not insert 'r8168': Exec format error
So, this machine was working fine, update, driver gone.