We show how to efficiently approximate any almost commuting $2$-local qubit Hamiltonian by a commuting one. As a consequence, we show that $\delta$-approximations to the ground energy for $\varepsilon$-almost commuting $2$-local $m$-term qubit Hamiltonians lie in $\mathsf{NP}$ when $\delta \gg m\varepsilon^{1/6}$, extending the classical containment well beyond the commuting setting. Additionally, we present two applications of our rounding framework: Gibbs sampling and fast Hamiltonian simulation for almost commuting systems.