Are those two distributions indistinguishable? The Decision composite residuosity problem problem states that is impossible to distinguish between those two ensembles:
$\{x^N \mod {N^2} | x \in \mathbb{Z^*_{N{^2}}}\}$ and $\{r  \in \mathbb{Z^*_{N{^2}}}\}$

Can we assume the same for these:
$\{x-1+x^N \mod {N^2} | x \in \mathbb{Z^*_{N{^2}}}\}$ and $\{r  \in \mathbb{Z^*_{N{^2}}}\}$
I see no reason to expect $x-1+x^N \bmod N^2$ to be indistinguishable from $r$, at least not based upon the assumption you give.  The map $f(x)= x^N \bmod N^2$ is a very different map from the map $g(x) = x-1+x^N \bmod N^2$.  The range of $f$ is a subgroup of size $\varphi(N)$; that's not true of $g$ (for instance, the range of $g$ can potentially be the entire group of integers modulo $N^2$).  The map $f$ is a $N$-regular map; I don't see any reason to expect $g$ to be $N$-regular in general, and maybe not even regular.  As a third difference, $f(x+kN) = f(x)$, but $g(x+kN) = g(x)+kN$.  For these reasons, I don't expect to find a simple reduction between these two indistinguishability statements.