The many-body-theory approach to positronium-atom interactions developed in [Phys. Rev. Lett. textbf{120}, 183402 (2018)] is applied to the sequence of noble-gas atoms He-Xe. The Dyson equation is solved separately for an electron and positron moving in the field of the atom, with the entire system enclosed in a hard-wall spherical cavity. The two-particle Dyson equation is solved to give the energies and wave functions of the Ps eigenstates in the cavity. From these, we determine the scattering phase shifts and cross sections, and values of the pickoff annihilation parameter $^1Z_text{eff}$ including short-range electron-positron correlations via vertex enhancement factors. Comparisons are made with available experimental data for elastic and momentum-transfer cross sections and $^1Z_text{eff}$. Values of $^1Z_text{eff}$ for He and Ne, previously reported in [Phys. Rev. Lett. textbf{120}, 183402 (2018)], are found to be in near-perfect agreement with experiment, and for Ar, Kr, and Xe within a factor of 1.2.
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