We consider the colored asymmetric simple exclusion process (ASEP) and stochastic six vertex (S6V) model with fully packed initial conditions; the states of these models can be encoded by 2-parameter height functions. We show under Kardar-Parisi-Zhang (KPZ) scaling of time, space, and fluctuations that these height functions converge to the Airy sheet. Several corollaries follow. (1) For ASEP and the S6V model under the basic coupling, we consider the 4-parameter height function at position $y$ and time $t$ with a step initial condition at position $x$ and time $s < t$, and prove that under KPZ scaling it converges to the directed landscape. (2) We prove that ASEPs under the basic coupling, with multiple general initial data, converge to KPZ fixed points coupled through the directed landscape. (3) We prove that the colored ASEP stationary measures converge to the stationary horizon. (4) We prove a strong form of decoupling for the colored ASEP height functions, as well as for the stationary two-point function, as broadly predicted by the theory of non-linear fluctuating hydrodynamics. The starting point for our Airy sheet convergence result is an embedding of these colored models into a larger structure -- a color-indexed family of coupled line ensembles with an explicit Gibbs property, i.e., a colored Hall-Littlewood line ensemble. The core of our work then becomes to develop a framework to analyze the edge scaling limit of these ensembles.