The parts of your body that generate heat and that can sense temperature and the loss of heat are insulated from the environment by a layer of dead skin cells. The total thermal conductivity to the environment is the thermal conductivity of the materials that you touch in series with the thermal conductivity of this layer of skin. Since this layer has a rather poor thermal conductivity itself, the sensation of touching different materials with much better thermal conductivity will not differ much. If this skin layer is broken, however, temperature and heat conductivity differences are felt much stronger, usually in a rather painful manner.
The heat flow (per unit area) through some thin layer, e.g. a boundary layer of water, is given by:
$$ \frac{dQ}{dt} = \frac{K\Delta T}{d} $$
where $K$ is the thermal conductivity, $d$ is the thickness of the layer and $\Delta T$ is the temperature difference between the two sides of the layer.
So a high thermal conductivity does indeed mean a high heat flow rate. But as your body loses heat, that heat goes into heating up the water. If the water had a low specific heat then it would heat up fast and you'd quickly be surrounded by a layer of water at your body temperature. This layer of water would then act as an insulator.
In the context of the equation above, a low specific heat means $\Delta T$ quickly reduces with time and that reduces the heat flow. Conversely, a high specfic heat means it takes a lot of the heat from your body to heat the water, and that tends to maintain $\Delta T$ at a high value.
So while you are quite correct that high thermal conductivity is a major factor in the heat loss to water, so is the high specific heat of the water.
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From a heat transfer perspective, steam is a lot more like air that it is like liquid water. The chemical makeup doesn't matter as much as the state.
As a gas, steam is a bunch of $H_2O$ molecules flying around at random, bashing into each other occasionally. The mechanism of conduction in that case is that the hot molecules will gradually bounce past the cold (and vice-versa), moving heat across a temperature gradient. The same is true for air, so their conductive properties are similar.
As a liquid, those molecules are much more tightly packed and are in contact (sort of) with each other all the time. When you heat up molecules in one place they can transfer that heat directly to their neighbors. At the molecular scale heat is just a lot of wiggling, spinning and vibrating motion (electrons get involved too if it's hot enough). Being in contact makes it easy for vibration in one molecule to make its way to others.
The pressure dependence in steam (an other gases) shows up at two extremes. At high pressure, the molecules will get closer together as they fly around and start to act a bit more like a liquid. At very low pressures, they'll collide so rarely that some of the molecules will go clear across the space in question without hitting any of the others. That the low pressure effect depends on both pressure and the size of the system