Methanol has some dehydration effect but normally it is not used for this purpose. It is mainly used for hydrate formation inhibition. By injecting methanol, it replaces water vapor in the gas phase due to its high vapor pressure. In effect, it pushes water vapor out of gas phase and sends it to the liquid phase. I did a quick checking using ProMax and found out the amount of methanol going to vapor phase is more than the amount of water leaving the gas phase.
The answer is that methanol has minimal dehydration capability and from a practical viewpoint it would be essentially zero. Methanol is used to inhibit hydrate formation by lowering the freezing point of liquid water that has already condensed. If you injected sufficient methanol into a gas to create a liquid methanol phase, methanol would lower the activity of water and reduce the vapor phase concentration somewhat, but the process would be horribly inefficient since methanol has a vapor pressure that is 3-4 times higher than the vapor pressure of water. Adsorbents should have very low vapor pressures to minimize vapor losses.
If you inject ethylene glycol into the gas, you also get minimal dewpoint suppression, but for a different reason. EG (MEG) has a very low vapor pressure and if it was injected at high purity would actually absorb water from the vapor phase, lowering the water dewpoint. But we do not inject highly concentrated EG into the gas. We typcially inject EG that is concentrated to about 80 wt%. The outlet EG concentration in pipelines is typically less than 50 wt%. At these concentrations the equilibrium dewpoint suppression is only a few degrees below the system temperature.
In both the case of methanol and EG injection, the water removal is caused by condensation (cooling the gas) and the inhibitor simply lowers the freezing (hydrate) temperature. Without the cooling, the water dewpoint suppression is neglible.