Normalized correlation coefficent

Given random variable $X \sim Normal(\mu_x, \sigma_x)$ and $Y = X + 1$, how can I calculate their normalized correlation coefficient?

I know the formulas:

$Corr(X, Y) = \frac{Cov(X, Y)}{\sqrt{\sigma_x^2 * \sigma_y^2}}$

$Cov(X, Y) = E[XY] - E[X]E[Y]$

In order to simplify the covariance calculation I need to establish whether the variables are independent or dependent. What can I say about their relationship? It seems like they are dependent given their definition, but if I look at their PDFs it seems like they are independent:

$X \sim Normal(\mu_x, \sigma_x)$

$Y \sim Normal(\mu_x + 1, \sigma_x)$

Considering $\mu_x$ is a constant value I can write $\mu_y = \mu_x + 1$, which will produce another constant, and I can rewrite Y:

$Y \sim Normal(\mu_y, \sigma_x)$

In essence I have 2 normally distributed random variables X and Y located at $\mu_x$ and $\mu_y$ respectively, which are independent. This logic seems right to me, but having chatted on #math I found strong disagreement without any explanation.

$$\mathbb{C}{\rm orr}[X,X + 1] = 1$$