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We validate the presence of a two-planet system orbiting the 0.15–1.4 Gyr K4 dwarf TOI 560 (HD 73583). The system consists of an inner moderately eccentric transiting mini-Neptune (TOI 560 b, $P={6.3980661}_{-0.0000097}^{+0.0000095}$ days, $e={0.294}_{-0.062}^{+0.13}$ , $M={0.94}_{-0.23}^{+0.31}{M}_{\mathrm{Nep}}$ ) initially discovered in the Sector 8 Transiting Exoplanet Survey Satellite (TESS) mission observations, and a transiting mini-Neptune (TOI 560 c, $P={18.8805}_{-0.0011}^{+0.0024}$ days, $M={1.32}_{-0.32}^{+0.29}{M}_{\mathrm{Nep}}$ ) discovered in the Sector 34 observations, in a rare near-1:3 orbital resonance. We utilize photometric data from TESS Spitzer, and ground-based follow-up observations to confirm the ephemerides and period of the transiting planets, vet false-positive scenarios, and detect the photoeccentric effect for TOI 560 b. We obtain follow-up spectroscopy and corresponding precise radial velocities (RVs) with the iSHELL spectrograph at the NASA Infrared Telescope Facility and the HIRES Spectrograph at Keck Observatory to validate the planetary nature of these signals, which we combine with published Planet Finder Spectrograph RVs from the Magellan Observatory. We detect the masses of both planets at >3 σ significance. We apply a Gaussian process (GP) model to the TESS light curves to place priors on a chromatic RV GP model to constrain the stellar activity of the TOI 560 host star, and confirm a strong wavelength dependence for the stellar activity demonstrating the ability of near-IR RVs to mitigate stellar activity for young K dwarfs. TOI 560 is a nearby moderately young multiplanet system with two planets suitable for atmospheric characterization with the James Webb Space Telescope and other upcoming missions. In particular, it will undergo six transit pairs separated by