Were they able to connect the wet steam from their cooking pots to water evaporation? I am assuming they would at least have understood the connection between heat and increased evaporation rates?
First, it is important to understand that the notion that things or states like gases exist did not come up before the 18th century. Only when Daniel Bernoulli wrote the Hydrodynamica in 1738, a theory of gas as a state different from liquid was established. Until then, the air was just deemed a very thin liquid, so the very question of the transformation of liquid water into steam did not arise in the modern sense - it was rather a question of transforming cold and heavy water into hot and light water.
In antiquity, Aristotle and other philosophers held on to the idea of 'natural places', i.e. that things with certain properties would by themselves move to their natural places - hot things would move up, cold things would sink down.
With the re-discovery and rise of atomic theory in the 17th century, people would reject this idea and resort to the idea of bodies being diluted until they can float up - the observation of buoyancy and different types of density makes it clear that a thick and heavy liquid such as water would flow down, while thin and diluted (i.e. gaseous) water would float up. Descartes discusses these phenomena in his Principles of Philosophy and Le Monde, for example.
(It's a bit more complicated, because the notion of density was partially rejected by people like Descartes and the Cartesians, so one would mostly assume that different types of weight are at stake here, but these explanations were obviously faulty and discussed as problematic, but it was difficult to come up with a more adequate notion of density before modern atomic theory and the discovery of the chemical elementary system.)
The theories of the ether also played a part here. The ether was often identified as both fire and heat and it was assigned the function of decreasing density - you heat up water and the ether flows from the fire into the water and decreases its weight by diluting the water. Take a look at Emily Du Châtelet's Dissertation on the Nature of Fire, for example.
I don't know anyone who would theorize on the relation between heat and evaporation / density, which to my understanding only comes up with modern thermodynamics.