Is there any liability upon me after I went back on selling my car to someone to whom I had said "I've sold it to you" out of embarrassment, then sold it to another person, knowing that the first buyer is demanding it from me?

A contract made over the phone has the same rulings as a contract made in person. A sale becomes binding upon the expiration of the option of the session (khiyar al-majlis), and in your case, this option has expired. As for mutual consent (al-taradhi), it is a condition for the validity of the sale. So, if you said, "I sell it to you," willingly, and then regretted it, the sale is complete, and your subsequent action of selling the car to another person is invalid. You must seek your friend's forgiveness. However, if you said it out of shyness and embarrassment, without genuine consent, then the sale is not binding upon you, and you are permitted to sell the car to someone else. In all cases, satisfying your friend and seeking his forgiveness is the best course of action.