We prove a semiclassical resolvent estimate for a broad class of non-self-adjoint, non-elliptic pseudodifferential operators in the low-lying spectral regime. The proof relies on improved ellipticity properties for the symbol of the operator, which we obtain by imposing a dynamical condition on the average of the real part of the symbol along the Hamiltonian flow generated by its imaginary part. An application of the resolvent estimate to a family of semiclassical Schr\"{ o}dinger operators with complex potentials allows us to localize the spectral problem to an $O(h)$-sized neighborhood of the origin.
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