We determine the Hausdorff, the packing and the box-counting dimensions of a family of self-affine sets generalizing Barański carpets. More specifically, we fix a Barański system and allow both vertical and horizontal random translations, while preserving the structure of the rows and columns. The alignment kept in the construction allows us to give expressions for these fractal dimensions outside of a small set of exceptional translations. Such formulae will coincide with those for the non-overlapping case, and thus provide examples where the box-counting and the Hausdorff dimension do not necessarily agree. These results rely on Hochman’s recent work on the dimensions of self-similar sets and measures, and can be seen as an extension of Fraser and Shmerkin [On the dimensions of a family of overlapping self-affine carpets. Ergod. Th. & Dynam. Sys.doi: 10.1017/etds.2015.21. Published online: 21 July 2015] results for Bedford–McMullen carpets with columns overlapping.