In doing a little unrelated carbonite freezing research, I came across mention of Roni von Wasaki, and was surprised to see mention of his use of the Jedi Mind Trick while encased in carbonite (emphasis mine):
Confiscating Wasaki's lightsaber and shoto, Drobo froze him in carbonite and stored him in the hold of his ship, the Starkiller.
Still attempting to resist capture in hibernation, Wasaki mentally reached out to Drobo, convincing the captain to free him and return his weapons. However, XT-8 learned of that and killed Drobo, as he had become a liability. In the process, XT-8 damaged the Starkiller's hyperdrive. The droid, unable to repair the ship, continued the trip on sublight drive, turning a thirty day jump into 20,000 years.
Wasaki persisted in his frozen state, continually trying to reach anyone passing within close proximity of the ship. Though the carbonite preserved him, the elongated journey slowly drove him insane, leaving him with twenty millennia of lonely thoughts and dreams. His situation also left him permanently blind, leaving him with only the Force to rely on.
Note that this is from an adventure for the Star Wars roleplaying game, making it potentially C-canon that yes, a Jedi could remain conscious while encased in carbonite, and influence the world (albeit perhaps not physical manipulation) from within the carbonite. However, the carbonite freezing process used on Wasaki was that of almost 20,000 years prior to the experimental technique used by Darth Vader on Han Solo, so it could be conceivable that some difference in the process resulted in later applications of carbon freezing inducing a true hibernation state.