We compute moments of $L$-functions associated to the polynomial family of Artin-Schreier covers over $\mathbb{F}_q$, where $q$ is a power of a prime $p$, when the size of the finite field is fixed and the genus of the family goes to infinity. More specifically, we compute the $k^{\text{th}}$ moment for a large range of values of $k$, depending on the sizes of $p$ and $q$. We also compute the second moment in absolute value of the polynomial family, obtaining an exact formula with a lower order term, and confirming the unitary symmetry type of the family. This is joint work with Alexandra Florea and Edna Jones.