Econ grad student here (and someone else converted away from Austrian econ in part from Caplan's article + debate with Block). Most of economics just chugs right along with the standard rationality (instrumental rationality, not epistemic) assumptions. Not because economists actually believe humans are rational - well some do, but I digress - but largely because we can actually get answers to real world problems out of the rationality assumptions, and sometimes (though not always) these answers correspond to reality. In short, rationality is a model and economists treat it as such - it's false, but it's an often useful approximation of reality. The same goes for always assuming we're in equilibrium. The trick is finding when and where the approximation isn't good enough and what your criteria for "good enough" is.

Now, this doesn't mean mainstream economists aren't interested in cogsci rationality. An entire subfield of economics - Behavioral Economics - rose up in tandem with the rise of the cogsci approach to studying human decision making. In fact, Kahneman won the nobel prize in economics. AFAICT there's a large market for economic research that applies behavioral economics to problems typically studied in classical, rational agent settings. The problem isn't the demand side - I think economists would love to see a fully general theory of general equilibrium with more plausible agents - it's the supply side: getting answers out of models with non-rational agents is a difficult task. It's already hard enough with rational agents for models to be anywhere near realistic - in macro models with micro foundations, we often assume all agents are identical and all firms are identical. This may seem terribly unrealistic, but often there's some other complication in the model that makes it hard enough to find solutions. Adding heterogenous firms and agents is an extra complication that may not add anything illuminating to the model. So, many economists treat the rationality assumptions which are fundamental to neoclassical economics similarly. If the rationality of agents within their model is tangential to the point they're trying to make (which may only be known empirically), they'll choose the easier assumption to work with. There are fields where the frailty of human rationality seems centrally important, and those are the fields where you're most likely to see nonstandard rationality assumptions. Behavioral Finance is an example of one of these.

The biggest thing I would say is, don't think in terms of "schools" of economic thought. Think in terms of models and tools. Most good ideas are eventually assimilated into the "neoclassical" economic toolkit in some form or another. And besides, thinking in terms of schools of thought is a good way to unintentionally mind-kill yourself.

As far as textbooks go, most higher level (intermediate micro and above) will present models without making any claims about when they're a good approximation and when they aren't. Oftentimes this is because the models being presented are actually just stepping stones to the more realistic and more complicated models economists are actually using. This is generally good, though I wish there were more empirical evidence presented. Any edition of Microeconomic Analysis by Varian will give you a good intermediate level (requires some calculus) rundown of standard micro theory. Think of it as taking standard economic intuitions (to economists - even austrians) and writing down equations that describe them so that we can talk about them precisely. I'd steer clear of any non-graduate level macro textbooks. The macro we teach undergrads is not the macro practicing macroeconomists actually believe. (Even on the graduate level, there isn't a generally accepted class of models that economist agree on, so it might not be that useful to study modern macro). If your mathematical background is stronger, Mas-Colell, Whinston and Green's Microeconomic Theory is a standard first year graduate micro text that's densely packed with a lot of material. Simon and Blume's Mathematics for Economists is the standard math primer used to prepare students for the class Mas-Colell is typically used in, if you're unsure about your math background.

Edit: Holy mother of grammar!