Would water stay on the moon without containment?

There is ice on the moon in deep craters which never get sunlight. Anywhere else and it would turn to steam and ‘escape into space, as the Moon’s low gravity cannot hold gas for any appreciable time’.

But we really don’t have a water shortage when you look at the planet as a whole. And what we use stays on Earth.

We do have localized shortages where it’s needed.

Desalinating and moving ocean water would be (I am guessing) orders of magnitude cheaper than transporting it from the Moon or beyond.