The Kepler mission revealed hundreds of systems with multiple transiting planets, which provide key insights into the correlations within planetary systems and their architectures if the detection biases are properly accounted for. He, Ford, & Ragozzine (2019, 2020) developed a forward model (SysSim) to perform statistical inference on the intrinsic distributions of planetary systems around FGK dwarfs, by modeling the Kepler detection pipeline and comparing simulated catalogs to the Kepler catalog. In those studies, we found that planetary systems are clustered in periods and in sizes, and the fraction of stars with planets (with Rp > 0.5REarth and 3d < P < 300d) increases towards later type stars as a linear function of Gaia bp-rp color. We also showed that the observed multiplicity distribution can be well produced by two populations consisting of a low and a high mutual inclination component (Kepler dichotomy). Here, we show that a broad, natural distribution of mutual inclinations arising from systems at the angular momentum deficit (AMD) stability limit can also match the observed population. In our new model, we distribute the maximum AMD of each multi-planet system amongst the planets and show that this results in a broad, multiplicity-dependent distribution of eccentricities and mutual inclinations. Systems with intrinsically more planets have lower eccentricities and mutual inclinations. For systems with more than five planets, their eccentricity and mutual inclination distributions are close to lognormal, unlike the previously assumed Rayleigh distributions. This trend with multiplicity arises from the dependence of the maximum system AMD on the minimum period ratio in the system, as systems with tightly-spaced planets must have low AMD in order to remain stable. We also find evidence that intrinsic single planets have higher eccentricities than multi-planet systems, although their distribution is not well constrained. We provide a public code for simulating (intrinsic and observed) planet catalogs from our models, which can be used to inform RV follow-up efforts in the search for additional non-transiting companions in systems with transiting planets (such as those observed from the TESS mission).