We present a novel approach to the problem of dual-functional radar and communication (DFRC) waveform design with adjustable peak-to-average power ratio (PAPR), while minimizing the multi-user communication interference and maintaining a similarity constraint towards a radar chirp signal. The approach is applicable to generic radar chirp signals and for different constellation sizes. We formulate the waveform design problem as a non convex optimization problem. As a solution, we adopt the alternating direction method of multipliers (ADMM), hence iterating towards a stable waveform for both radar and communication purposes. Additionally, we prove convergence of the proposed method and analyze its computational complexity. Moreover, we offer an extended version of the method to cope with imperfect channel state information (CSI). Finally, we demonstrate its superior performance through simulations, in comparison to state-of-the-art radar-communication waveform designs.