We introduce a new interacting particle system on $\mathbb{Z}$, \emph{slowed $t$-TASEP}. It may be viewed as a $q$-TASEP with additional position-dependent slowing of jump rates depending on a parameter $t$, which leads to discrete and nonuniversal asymptotics at large time. If on the other hand $t \to 1$ as $\text{time} \to \infty$, we prove (1) a law of large numbers for particle positions, (2) a central limit theorem, with convergence to the fixed-time Gaussian marginal of a stationary solution to SDEs derived from the particle jump rates, and (3) a bulk limit to a certain explicit stationary Gaussian process on $\mathbb{R}$, with scaling exponents characteristic of the Edwards-Wilkinson universality class in $(1+1)$ dimensions. The proofs relate slowed $t$-TASEP to a certain Hall-Littlewood process, and use contour integral formulas for observables of this process. Unlike most previously studied Macdonald processes, this one involves only local interactions, resulting in asymptotics characteristic of $(1+1)$-dimensional rather than $(2+1)$-dimensional systems.