Improving the use of Type Ia supernovae (SNIa) as standard candles requires a better approach to incorporate the relationship between SNIa and the properties of their host galaxies. Using a spectroscopically-confirmed sample of $sim$1600 SNIa, we develop the first empirical model of underlying populations for SNIa light-curve properties that includes their dependence on host-galaxy stellar mass. These populations are important inputs to simulations that are used to model selection effects and correct distance biases within the BEAMS with Bias Correction (BBC) framework. Here we improve BBC to also account for SNIa-host correlations, and we validate this technique on simulated data samples. We recover the input relationship between SNIa luminosity and host-galaxy stellar mass (the mass step, $gamma$) to within 0.004 mags, which is a factor of 5 improvement over the previous method that results in a $gamma$-bias of ${sim}0.02$. We adapt BBC for a novel dust-based model of intrinsic brightness variations, which results in a greatly reduced mass step for data ($gamma = 0.017 pm 0.008$), and for simulations ($gamma =0.006 pm 0.007$). Analysing simulated SNIa, the biases on the dark energy equation-of-state, $w$, vary from $Delta w = 0.006(5)$ to $0.010(5)$ with our new BBC method; these biases are significantly smaller than the $0.02(5)$ $w$-bias using previous BBC methods that ignore SNIa-host correlations.
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