One way to obtain the -2/3 is by singularity analysis.

The first step is to construct the generating function of your sequence.
From the Taylor expansion of the Lambert $W$ function at 0, one gets that $-W(-x)$ is the generating function of the sequence $N^{N-1}/N!$ and therefore by differentiation
$$\frac{-W(-x)}{1+W(-x)}=\sum_{N\ge1}{\frac{N^N}{N!}x^N}.$$
Replacing $x$ by $x/e$ and multiplying by $1/(1-x)$ yields the desired generating function
$$F(x):=\frac{-W(-x/e)}{(1-x)(1+W(-x/e))}=
\sum_{N\ge1}{\left(\sum_{n=1}^N{\frac{n^ne^{-n}}{n!}}\right)x^N}.$$

From there, the result follows from an analysis at $x=1$. From the known expansion of $-W(-x)$ at $1/e$, one deduces
$$F(x)=\frac{\sqrt{2}}{2(1-x)^{3/2}}-\frac{2}{3(1-x)}+O\left(\frac1{\sqrt{1-x}}\right),\quad x\rightarrow1.$$

Now singularity analysis (or Darboux's method) deduces the asymptotic expansion of your sequence as
$$\frac{\sqrt{2 N}}{\sqrt{\pi}}-\frac23+O(1/\sqrt{N}).$$

With slightly more work along the same lines, one obtains a full asymptotic expansion beginning with
$${\frac {\sqrt {2N}}{\sqrt {\pi}}}-\frac23+{\frac {
\sqrt {2}}{3\sqrt {\pi N}}}-{\frac {37\sqrt {2}}{864\sqrt {\pi}N^{3/2}}}+{\frac {359\sqrt {2}
}{64800\sqrt {\pi}N^{5/2}}}+O \left( {N}^{-7/2} \right)
.$$