I consulted with another graduate student friend of mine who is taking a meteorology (weather) class, and he said that the average water content of the atmosphere is about 2 grams per kg, or 1/5 of a percent of the mass of the atmosphere. That being the case, it is unlikely that humidity variation will have a great effect. On the other hand, the natural pressure variation that we tend to see do to weather changes is less than a percent (I am remembering variation between about 29.8 and 30.0 inches of mercury, which is a little over a 0.6% variation), so the added effect due to moisture could in fact be a sizable, although not dominant, factor in the pressure equation!
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I believe wet air would in fact be heavier and would thus add to the air pressure, but I don't think the effect would be very great. The difference between dry air (0% humidity) and saturated air (100% humidity), if a friend of mine did the calculations correctly, would produce about a change equivalent to about a third of the difference between commonly observed extremes in barametric pressure. In other words, humidity will play a part but cannot account for the fluctuations
that we see.

Relative humidity is the partial pressure of water in the air divided by the vapor pressure of water at the air temperature. Vapor pressure is the pressure at which liquid is in equilibrium with it's gaseous state; water at a given temperature boils when the partial pressure of water is at or below its vapor pressure (at 100 degrees Celsius, the vapor pressure of water is one atmosphere). The vapor pressure of water is relatively low at room temperature, less than 0.1 atmosphere. So, humidity (as defined) really doesn't seem to depend on the air pressure. Air at high pressure or lower pressure with the same humidity and temperature would have the same amount of water in it.

I don't know much about weather, so I can't really say much about the relationship between air pressure and humidity. As I recall though, high pressure seems to keep storms and moisture away whereas storms seem to move into low pressure areas. This would seem to explain your discovery that humidity in inversely proportional to air pressure. If I can come up with a better answer I'll let you know.
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I think a better way to think about relative humidity is that it is the ratio between the amount of water in the air and the amount of water that can be stored in the air. 100% humidity would mean that the air is holding as much water as it can. The amount of water that can be in air depends on temperature. Colder air can hold less water than warm air.

It turns out that in a high pressure area, air is flowing out to areas of lower pressure. This air is replaced by air from the upper atmosphere which is colder and thus drier. So people inside a high pressure area notice that the air is drier than usual. Likewise, low pressure areas draw in warmer, moist air so people in a low pressure area would notice higher humidity.

Storms occur when a cold air mass and a warm air mass meet. Since the warm air and cold air don't mix very well, one air mass rises and the other doesn't. Which air mass rises? When this air mass rises to high altitude what do you think happens to its temperature? How would this effect its humidity? What happens if the humidity become higher than 100%?

By the way, I made a mistake before when I was talking about vapor pressure. A liquid boils at a temperature at which the vapor pressure equals the total pressure over the liquid. Thus water boils at 100 Celsius because the vapor pressure of water at that temperature is one atmosphere. Sorry about any confusion I caused!

This is one of the things that go somewhat against general experience. Humidity is defined to be:

actual partial pressure of water in air at a given temperature

divided by partial pressure of water in air in equilibrium with liquid water at that temperature.

Another way of saying this is that humidity is the mole fraction of water vapor in the air divided by the mole fraction of water vapor in air in equilibrium with liquid water.

During a rainstorm, or a heavy fog, liquid drops are suspended in the air (for longer or shorter times) and the air is fully saturated with water vapor in equilibrium with the water drops at that temperature. This is 100% humidity.

There can only be so much water vapor in air at a given temperature, and this amount increases very quickly with increasing temperature. So 100% humidity means different mole fractions of water vapor in the air at different temperatures. Generally, when an atmosphere at 100% humidity at a high temperature is cooled to a lower temperature, liquid water condenses - that is, it rains.

However, wet air is actually less dense than dry air. If you think about air as an ideal gas, there is a simple relationship between the density and the average molecular weight of the air. Since the molecular weight of air is about 29 gr/mol and water is 18 gr/mol, increasing the fraction of water in the air decreases the average molecular weight, and so decreases the density.

Formally:

(Pressure x Molecular Weight)
divided by
(Temperature x Gas Constant) = density

So wet air is less dense than dry air, until it starts to condense.