  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point-like sources and hence are now superceded.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.
