Angular and spectral differential imaging is an observational technique of choice to investigate the immediate vicinity of stars. By leveraging the relative angular motion and spectral scaling between on-axis and off-axis sources, post-processing techniques can separate residual star light from light emitted by surrounding objects such as circumstellar disks or point-like objects. This paper introduces a new algorithm that jointly unmixes these components and deconvolves disk images. The proposed algorithm is based on a statistical model of the residual star light, accounting for its spatial and spectral correlations. These correlations are crucial yet remain inadequately modeled by existing reconstruction algorithms. We employ dedicated shrinkage techniques to estimate the large number of parameters of our correlation model in a data-driven fashion. We show that the resulting separable model of the spatial and spectral covariances captures very accurately the star light, enabling its efficient suppression. We apply our method to datasets from the VLT/SPHERE instrument and compare its performance with standard algorithms (median subtraction, PCA, PACO). We demonstrate that considering the multiple correlations within the data significantly improves reconstruction quality, resulting in better preservation of both disk morphology and photometry. With its unique joint spectral modeling, the proposed algorithm can reconstruct disks with circular symmetry (e.g., rings, spirals) at intensities one million times fainter than the star, without needing additional reference datasets free from off-axis objects.