We show that the counts of low degree irreducible factors of a random polynomial $f$ over $\mathbb{F}_q$ with independent but non-uniform coefficients behave like that of a uniform random polynomial, exhibiting a form of universality for random polynomials over finite fields. Our strongest results require various assumptions on the parameters, but we are able to obtain results requiring only $q=p$ a prime with $p\leq \exp({n^{1/13}})$ where $n$ is the degree of the polynomial. Our proofs use Fourier analysis, and rely on tools recently applied by Breuillard and Varjú to study the $ax+b$ process, which show equidistribution for $f(α)$ at a single point. We extend this to handle multiple roots and the Hasse derivatives of $f$, which allow us to study the irreducible factors with multiplicity.