This paper presents a novel framework for enhancing the security, data rate, and sensing performance of integrated sensing and communications (ISAC) systems. We employ a random frequency and pulse repetition interval (PRI) agility (RFPA) method for the waveform design, where the necessary random sequences are governed by shared secrets. These secrets, which can be pre-shared or generated via channel reciprocity, obfuscate critical radar parameters like Doppler frequency and pulse start times, thereby significantly impeding the ability to perform reconnaissance from a passive adversary without the secret key. To further introduce enhanced data throughput, we also introduce a hybrid information embedding scheme that integrates amplitude shift keying (ASK), phase shift keying (PSK), index modulation (IM), and spatial modulation (SM), for which a low-complexity sparse-matched filter receiver is proposed for accurate decoding with practical complexity. Finally, the excellent range-velocity resolution and clutter suppression of the proposed waveform are analyzed via the ambiguity function (AF).