We consider the dimensions of a family of self-affine sets related to the Bedford-McMullen carpets. In particular, we fix a Bedford-McMullen system and then randomize the translation vectors with the stipulation that the column structure is preserved. As such, we maintain one of the key features in the Bedford-McMullen set-up in that alignment causes the dimensions to drop from the affinity dimension. We compute the Hausdorff, packing and box dimensions outside of a small set of exceptional translations, and also for some explicit translations even in the presence of overlapping. Our results rely on, and can be seen as a partial extension of, Hochman's recent work on the dimensions of self-similar sets and measures.