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Angular and spectral differential imaging is an observational technique of choice to investigate the immediate vicinity of stars. The relative angular motion and spectral scaling between on-axis and off-axis sources are exploited by post-processing techniques to separate two components: the residual star light and the light coming from surrounding objects such as circumstellar disks or point-like objects. This paper introduces a new algorithm to jointly unmix these two components and deconvolve disk images. The proposed algorithm is based on a statistical modeling of the residual star light, accounting for its spatial and spectral correlations. While critical, these correlations are not modeled and compensated for by existing reconstruction algorithms. We leverage dedicated shrinkage techniques to estimate the large number of parameters of our model of correlations in a data-driven fashion. We show that the resulting separable model of the spatial and spectral covariances captures very accurately the star light, thus allowing its efficient suppression. We apply our method on several datasets from the VLT/SPHERE instrument and compare the performances against standard algorithms of the field (cADI, PCA). We show that accounting for the multiple correlations of the data significantly enhances the reconstruction quality, leading to better preservation of both the disk morphology and photometry. Thanks to its unique joint spectral modeling of the data, the proposed algorithm can reconstruct disks having a circular symmetry (e.g., rings, spirals) at intensities one million times fainter than the star, without resorting to additional reference datasets free from the off-axis objects.