Linear functions on the classical matrix groups

Let $M$ be a random matrix in the orthogonal group $\mathcal{O}_n$, distributed according to Haar measure, and let $A$ be a fixed $n\times n$ matrix over $\mathbb{R}$ such that $\mathrm{Tr}(AA^t)=n$. Then the total variation distance of the random variable $\mathrm{Tr}(AM)$ to a standard normal random variable is bounded by $\frac{2\sqrt{3}} {n-1}$, and this rate is sharp up to the constant. Analogous results are obtained for $M$ a random unitary matrix and $A$ a fixed $n\times n$ matrix over $\mathbb{C}$. The proofs are applications of a new abstract normal approximation theorem which extends Stein's method of exchangeable pairs to situations in which continuous symmetries are present.