Moist Chlorine & Hydrogen Psychrometric or Mollier Chart
I have searched hard but without proper psychrometric data am unable to even estimate preliminary values. 


You have to google for 'vapor pressure' of water and ideal gas model. In cases ideal gas model is close enough to true, the total pressure is divided into partial pressures, attributed to different gases, and the partial pressures are proportional to amount of molecules in the gas mixture. So, in the case the equilibrium partial pressure of water vapors above the liquid can be found in any table of vapor pressures, the rest of the pressure would be other gases. In the case of vapor above saline water, the equilibrium pressure is reduced by the fact that the salt is present. The exact reduction is guarded by Raoult's Law and is linearly proportional to the concentration of the salt. 


  Two coolers (shell/tube exchanger with chilled water on shell side) are planned to dehumidify these two gases.


To dehumidify these gases effectively, if this is really required, consider bubbling them through layer of concentrated $\ce{H2SO4}$, it is a quite common and easy way to dehumidify gases.