Chọn phương án D.

Giả sử \(z=a+b\mathrm{i}\) (\(a,\,b\in\mathbb{R}\)).

Suy ra \(\overline{z}=a-b\mathrm{i}\).

Khi đó: $$\begin{aligned}
&\,z+2\overline{z}&=2+3\mathrm{i}\\
\Leftrightarrow&\,a+b\mathrm{i}+2(a-b\mathrm{i})&=2+3\mathrm{i}\\
\Leftrightarrow&\,3a-b\mathrm{i}&=2+3\mathrm{i}\\
\Leftrightarrow&\begin{cases}3a&=2\\ -b&=3\end{cases}\\
\Leftrightarrow&\begin{cases}a&=\dfrac{2}{3}\\ b&=-3\end{cases}
\end{aligned}$$
Vậy \(z=\dfrac{2}{3}-3\mathrm{i}\).

Suy ra \(|z|=\sqrt{\left(\dfrac{2}{3}\right)^2+(-3)^2}=\dfrac{\sqrt{85}}{3}\).