Distribution of angle between two dependent gaussian random vectors

Question

Suppose that $x,y \in \mathbb{R}^n$ have i.i.d. $\mathcal{N}(0,1)$ entries. For some scalars $\alpha,\beta \in [-1,1]$, I am interested in the distribution of the angle $\theta$ between $x$ and $\alpha x + \beta y$. With the dependence between $x$ and $\alpha x + \beta y$, I am unsure how to go about this. Ultimately, I would like to compute something of the form $\mathbb{E}[g(\theta)]$ where $g$ is a continuous function. Any help would be appreciated!