Which is denser, warm moist air at a given pressure, or warm dry air at the same pressure?

It's an interesting question, because it involves the gas law. Note that the question simplifies the gas law by removing one of the variables. Which variable is made into a constant by the constraints of the question?

It turns out that gas constant is often defined for an air parcel with NO water vapor. We've been treating the gas constant as just that, a constant.

What impact on the density of air parcel would adding water vapor have? Or, to put it another way, suppose there were two air parcels, one with a temperature of 97F and a dew point of 15F and another with the same temperature of 97F but a dew point of 70F. Which do you think is denser?

It turns out that water vapor molecules are very "bulky" for their weight. In other words, a gram of water vapor takes up much more space than a gram of diatomic oxygen or diatomic nitrogen. That should give you a clue

Virtual Temperature

To make the Ideal Gas Law as we've discussed it work, we can define a temperature, Virtual Temperature, that works this way. The Gas Law is very complicated when you consider the effects of the density of water vapor. But one can (not you, but one can) solve the more complicated Gas Law for density (including water vapor) and then put that density back into the familiar Gas Law and solve for temperature. That temperature is called Virtual Temperature. In other words, the impact of water vapor is to make a kilogram air parcel LESS dense. The virtual temperature of a moist air parcel is the temperature at which a theoretical dry air parcel would have a total pressure and density equal to the moist parcel of air. As long as there is a dew point (water vapor is present), the Virtual Temperature is always greater than the actual temperature.

Try it. Here's a simple online virtual temperature calculator.