As the astronauts observed, it first boils and then slowly freezes. If you put water in space, the water would start to boil on its own due to the fact that the vapor pressure would exceed the pressure of the environment the water is in. And, after that, it would start to crystallize.
However, the reason why it would freeze is not mainly the vaporizing water's absorption of heat from its surroundings. As the pressure decreases, the latent enthalpy of formation of water increases which would support you point that the vaporizing water would absorb even more heat from the remaining water; and therefore, it would decrease the temperature of remaining water to a temperature where desublimation(phase transition from gas to solid) takes place. However, since while freezing the water, it would have to give heat, the temperature of surrounding environment would increase which would result in sublimation of that small amount of frozen water.
However, since the surrounding environment would be space, the vaporized water would slowly lose heat by means of radiation and its temperature would decrease to desublimation temperature. Therefore, it would start to crystallize and what we are left with would be crystals of frozen water.
To understand this better you can look at the phase diagram of water below:
