We develop a method for bounding the sensitivity of solutions to stochastic differential equations (SDEs) to changes in the drift, $F$, and diffusion, $σ$, by using a combination of information-theoretic uncertainty quantification bounds, functional inequalities, and judiciously chosen coupled auxiliary SDEs. The method is capable of producing non-asymptotic bounds which are well behaved in the $T\to \infty$ limit and does not require the perturbations to $F$ and $σ$ to be small. Our approach applies to expectations of both time-averaged and exponentially discounted observables and also produces sensitivity bounds for linear parabolic PDEs. When applied to stationary solutions and Lipschitz observables, our results produce bounds on the $1$-Wasserstein distance between invariant measures which have optimal scaling in each error term. The present method significantly expands on prior information-theoretic SDE sensitivity bounds, which are only applicable to perturbations of the drift.