Question on the definition of one-sided PSD in AWGN channel
The relation between one-sided and two-sided noise PSD is correct. But note that the power of a complex random variable $n$ is given by $E\{|n|^2\}$, and not by $E\{n^2\}$. So in your case you obtain $E\{|n|^2\}=\sigma^2_x+\sigma^2_y$.