  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 Varj\'u to study the $ax+b$
process, which show equidistribution for $f(\alpha)$ 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.
