  In this paper, we study two classes of planar self-similar fractals
$T_\varepsilon$ with a shifting parameter $\varepsilon$. The first one is a
class of self-similar tiles by shifting $x$-coordinates of some digits. We give
a detailed discussion on the disk-likeness ({\it i.e., the property of being a
topological disk}) in terms of $\varepsilon$. We also prove that
$T_\varepsilon$ determines a quasi-periodic tiling if and only if $\varepsilon$
is rational. The second one is a class of self-similar sets by shifting
diagonal digits. We give a necessary and sufficient condition for
$T_\varepsilon$ to be connected.
