We propose the affine filter bank modulation (AFBM) waveform for enhanced integrated sensing and communications (ISAC) in sixth generation (6G), designed by drawing on concepts from classical filter bank multicarrier modulation (FBMC) theory and recent advances in chirp-domain waveforms, particularly affine frequency division multiplexing (AFDM). Specifically, AFBM exhibits several desirable properties, with emphasis on its remarkably low peak-to-average power ratio (PAPR) and reduced out-of-band emission (OOBE) when benchmarked against the conventional AFDM waveform under doubly-dispersive (DD) channel conditions. In the communications setting, reliable symbol detection is achieved using a tailored low-complexity Gaussian belief propagation (GaBP)-based algorithm, while in the sensing setting, a range and velocity estimation approach is developed that integrates an expectation maximization (EM)-assisted probabilistic data association (PDA) framework to accurately identify surrounding targets. The highlighted performance and benefits of AFBM are validated through analytical and numerical evaluations, including conventional metrics such as ambiguity function (AF), bit error rate (BER), and root mean square error (RMSE), consolidating its position as a promising waveform for next-generation wireless systems.