Persistent excesses in the spectra of gamma rays from the Galactic Center and cosmic ray antiprotons can be explained by dark matter of mass 50-65 GeV annihilating into b quarks, but this is typically hard to reconcile with direct detection constraints. We resolve this tension using a simple class of models, where dark matter is a pseudo-Nambu-Goldstone boson, having naturally momentum-suppressed couplings to nuclei. Exploring the parameter space of the model, we find that it can explain the cosmic ray anomalies while remaining compatible with constraints from the relic abundance and annihilation in dwarf spheroidal galaxies. In certain regions of parameter space, the Higgs-dark matter coupling can help stabilize the Higgs potential up to the Planck scale. The scalar partner of the dark matter is an extra Higgs boson, that can explain a tentative diphoton excess observed by CMS, and an excess of b (b) over bar signal from LEP, if its mass is similar to 96 GeV and the model is extended to include a heavy scalar quark. This extended model predicts a monochromatic gamma-ray line near 64 GeV, at a level close to current experimental sensitivity, from dark matter annihilations in the Galaxy.