Traditional dark matter (DM) models, eg. WIMPs, assume dark matter is weakly coupled to the standard model so that elastic scattering between DM and baryons can be described perturbatively by Born approximation. Most direct detection experiments are analyzed according to that assumption. We show that when the interaction is attractive and strong, DM-nucleus scattering exhibits rich resonant behavior with a highly non-trivial dependence on atomic mass. The scattering is non-perturbative in much of the natural parameter range, and a full numerical calculation is needed. We also show that the extended rather than point-like nature of nuclei significantly impacts the cross sections and must therefore be properly taken into account. These effects are particularly important for dark matter with GeV-scale masses, near the boundary of exclusion regions from existing direct detection limits. They also affect the interpretation of CMB constraints, as we show. We report the corrected limits, which are in some respects weaker and in other respects stronger than previous bounds in the literature, which were based on perturbation theory and point-like sources and hence are now superceded. Sexaquark ($uuddss$) DM with mass $lesssim 2$ GeV, which exchanges QCD mesons with baryons, remains unconstrained for most of the parameter space of interest.
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