We present a quantitative model of Galactic cosmic-ray (GCR) origin and acceleration, wherein a mixture of interstellar and/or circumstellar gas and dust is accelerated by a supernova remnant blast wave. The gas and dust are accelerated simultaneously, but differences in how each component is treated by the shock leave a distinctive signature, which we believe exists in the cosmic-ray composition data. A reexamination of the detailed GCR elemental composition, presented in a companion paper, has led us to abandon the long-held assumption that GCR abundances are somehow determined by first ionization potential. Instead, volatility and mass (presumably mass-to-charge ratio) seem to better organize the data: among the volatile elements, the abundance enhancements relative to solar increase with mass (except for the slightly high H/He ratio); the more refractory elements seem systematically overabundant relative to the more volatile ones in a quasi--mass-independent fashion. If this is the case, material locked in grains in the interstellar medium must be accelerated to cosmic-ray energies more efficiently than interstellar gas-phase ions. Here we present results from a nonlinear shock model that includes (1) the direct acceleration of interstellar gas-phase ions, (2) a simplified model for the direct acceleration of weakly charged grains to ~100 keV amu-1 energies, simultaneously with the acceleration of the gas ions, (3) the energy losses of grains colliding with the ambient gas, (4) the sputtering of grains, and (5) the simultaneous acceleration of the sputtered ions to GeV and TeV energies. We show that the model produces GCR source abundance enhancements of the volatile, gas-phase elements that are an increasing function of mass, as well as a net, mass-independent enhancement of the refractory, grain elements over protons, consistent with cosmic-ray observations. We also investigate the implications of the slightly high H/He ratio. The GCR 22Ne excess may also be accounted for in terms of the acceleration of 22Ne-enriched presupernova Wolf-Rayet star wind material surrounding the most massive supernovae. We also show that cosmic-ray source spectra, at least below ~1014 eV, are well matched by the model.