We propose a high-spectral-efficiency multicarrier system for communication over the doubly dispersive (DD) channel which yields very low frame error rate (FER), with quadratic (in the frame length) receiver complexity. To accomplish this, we combine a non-(bi)orthogonal multicarrier modulation (MCM) scheme recently proposed by the authors with novel sequence detection (SD) and channel estimation (CE) algorithms. In particular, our MCM scheme allows us to accurately represent the DD channels otherwise complicated intercarrier interference (ICI) and intersymbol interference (ISI) response with a relatively small number of coefficients. The SD and CE algorithms then leverage this sparse ICI/ISI structure for low-complexity operation. Our SD algorithm combines a novel adaptive breadth-first search procedure with a new fast MMSE-GDFE preprocessor, while our CE algorithm uses a rank-reduced pilot-aided Wiener technique to estimate only the significant ICI/ISI coefficients.