We present a scheme for correction of x-y-separable aberrations in optical coherence tomography (OCT) designed to work with phase unstable systems with no hardware modifications. Our approach, termed SHARP, is based on computational adaptive optics and numerical phase correction and follows from the fact that local phase stability is sufficient for the deconvolution of optical aberrations. We demonstrate its applicability in a raster-scan polygon-laser OCT system with strong phase-jitter noise, achieving successful refocusing at depths up to 4 times the Rayleigh range. We also present in vivo endoscopic and ex vivo anterior segment OCT data, showing significant enhancement of image quality, particularly when combining SHARP results with a resolution-preserving despeckling technique like TNode.