Graphite: Thermal and electrical conductivity
You are wrong to say that graphite has free electrons.  The electrons are bound to sheets of aromatic rings--delocalized but not free. In electrical conduction, the electron pairs are able to jump from bond to bond on the sheet and between sheets. It is worth noting that graphite has anisotropic conduction where between sheets conduction is lower than conduction along a sheet.

For thermal conduction, energizing an atom does not transmit the energy via free electrons like a metal would.  instead, you are limited to a predominately phonon driven heat transfer again, but now the structure is less dense and the carbons are less rigidly bonded.

Diamond as your correctly identified has a rigid structure that can easily conduct phonons but not electrons which gives good thermal, but not electrical conduction.