Flow resistance of liquids flowing through microchannels can be reduced by replacing flat, no-slip boundaries with boundaries adjacent to longitudinal grooves containing an inert gas, resulting in apparent slip. With applications of such textured microchannels in areas such as microfluidic systems and direct liquid cooling of microelectronics, there is a need for predictive mathematical models that can be used for design and optimization. In this work, we describe a model that incorporates the physical effects of gas viscosity (interfacial shear), meniscus protrusion (into the grooves), and channel aspect ratio and show how to generate accurate solutions for the laminar flow field using Chebyshev collocation and domain decomposition numerical methods. While the coupling of these effects are often omitted from other models, we show that it plays a significant role in the behavior of such flows. We find that, for example, the presence of gas viscosity may cause meniscus protrusion to have a more negative impact on the flow rate than previously appreciated. Indeed, we show that there are channel geometries for which meniscus protrusion increases the flow rate in the absence of gas viscosity and decreases it in the presence of gas viscosity. In this work, we choose a particular definition of channel height: the distance from the base of one groove to the base of the opposite groove. Practically, such channels are used in constrained geometries and therefore are of prescribed heights consistent with this definition. This choice allows us to easily make meaningful comparisons between textured channels and no-slip channels occupying the same space.