Is it possible to store moist air at high pressure?
There is one importing principle to consider.
Air does not store vapor. Space does. Less space means less vapor capacity, no matter how much of air is there. (*)
If air is being compressed, the partial vapor pressure increases. It it reaches the saturated vapor pressure at given temperature, water vapor starts to condensate. When air is then expanded, its  absolute and relative humidity decreases, according to the expansion ratio.
E.g. Lets have 1 cubic metre of compressed air with 100% relative humidity. If it expands to 2 cubic metres and temperature raises back to the initial temperature, its relative humidity will be 50 %.
So compression provides liquid water and expansion needs liquid water to restore humidity.
If there is compressed air with cold water and if pressure drop e.g. 20 to 1 atm) then its relative humidity suddenly drops from 100% to 5% (when air warms up back after adiabatic cooling down.). Water exposed to low humidity air will start slow evaporation.
It will not help much in humidity restoration, unless some active evaporation technique is involved a/o the air release is slow, passing evaporation unit.

(*) Exactly speaking, saturated vapor pressure and therefore vapor capacity of space very slighly increases with total pressure. But it is negligible in context of your question.

Note that absolute humidity is in g/m3 - it is intensive, not extensive variable. Rather, the vapour content  = absolute humidity * volume.