Ok, so I asked about this on a physics forum, and here's my (medium-confidence) interpretation of the information that I gleaned:
1) If the planet has oceans in proportions roughly the same as earth's, the diffusion of heat through the ocean might be enough to prevent the ocean on the night side from freezing, although the landmasses would probably be mostly ice. This would solve the problem of water scarcity--in fact, it makes it quite probable that
atmospheric water would be abundant, since the ocean in the middle of the day side would always be boiling/near boiling.
2) If the planet's oceans are too small to be self-sustaining in distributing heat, the water would end up frozen on the night side--but there would still always be a small amount of water in the atmosphere. Ice sublimes even at -133 degrees celsius, so there would be water vapor in the air--and the difference in temperature between the day and night sides would cause a huge
Hadley cell to form around the terminator, circulating dry/moist air between the day and night sides.
It would be a desert world for certain, but pockets of life might exist where mountains and other geological features cause water to condense in the twilight belt. People could collect water using
wind traps like in Frank Herbert's
Dune, or by directly mining ice on the night side.
If either of these options works for your world setting, perhaps you could get by without the need for climate-sustaining cobha.