As hot air in the atmosphere rises it expands and cools at the same time. If the air cools though, shouldn't its density increase?
No, its density does not need to increase just because it is cooling. Consider, for example, gas in an insulated cylinder after rapid compression: at first, it is quite hot, but then since no insulator is perfect, it slowly loses heat, lowering temperature at the same time. But it has the same gas particles in the same volume, so it has no chance to change density. Yet it is cooling and the pressure is dropping too.
This illustrates why you need to consider, for each thermodynamics problem, whether conditions are adiabatic, isobaric, isentropic, isothermal, isoenthalpic, isochoric or (worse yet), none of the above. Now which of the above do you think your hot air example is? It is not reversible, is it?