Hi,
Looking closely, I can see that the clouds are changing, it is just hard to tell that as they are moving quite fast.
I would suggest the following:
• Freeze the clouds (no offset animation)
• Play around Phase and Variety parameters, understand what values give you the desired effect with desired amount. You may also want to try animating the amount as well
• Only after getting the Phase/Variety/Shape changes you want, try animating the movement.
For seemingly moving only on one axis: both axes work, for sure. You may just need to increase the value for Offset, clouds have pretty large scale and values like 3km can appear to not have much effect.
If you think it make sense to have change clouds offset unit from "m" to "km", feel free to mention about it. I have often found that too large values should be used as "meters" for clouds to move the amount wanted.