  We determine the Hausdorff, packing and box-counting dimension of a family of
self-affine sets generalizing Bara\'nski carpets. More specifically, we fix a
Bara\'nski system and allow both vertical and horizontal random translations,
while preserving the rows and columns structure. The alignment kept in the
construction allows us to give expressions for these fractal dimensions outside
of a small set of exceptional translations. Such formulas will coincide with
those for the non-overlapping case, and thus provide examples where the
box-counting and Hausdorff dimension do not necessarily agree. These results
rely on M. Hochman's recent work on the dimensions of self-similar sets and
measures, and can be seen as an extension of J. Fraser and P. Shmerkin results
for Bedford-McMullen carpets with columns overlapping.
