Yes, True and False have to be omniscient to be able to answer consistently correctly or incorrectly, for any arbitrary binary question. There's a version of the answer which (spoiler) relies on asking unanswerable questions, which only Random would answer. There's also solution that doesn't rely on such gimmicks, however.
There's nothing in your wording that suggests random is not able to refuse an unanswerable question as one of it's potential random responses.
P/S/A: There are single sentences which can create life-changing amounts of difference.