  We study the Assouad and quasi-Assoaud dimensions of dominated rectangular
self-affine sets in the plane. In contrast to previous work on the dimension
theory of self-affine sets, we assume that the sets satisfy certain separation
conditions on the projection to the principal axis, but otherwise have
arbitrary overlaps in the plane. We introduce and study regularity properties
of a certain symbolic non-autonomous iterated function system corresponding to
"symbolic slices" of the self-affine set. We then establish dimensional
formulas for the self-affine sets in terms of the dimension of the projection
along with the maximal dimension of slices orthogonal to the projection. Our
results are new even in the case when the self-affine set satisfies the strong
separation condition: in fact, as an application, we show that self-affine sets
satisfying the strong separation condition can have distinct Assouad and
quasi-Assouad dimensions, answering a question of the first named author.
