Why do you think the temperature does not change much during a phase change?

It depends on where the temperature is dropping, in which body so to speak. Generally, the temperature adapts to the two bodies, for example if a hot piece of metal meets a cold one, the two will continue until they are at an equal temperature, an intermediate temperature.

By the there is no change in the temperature until the phase change is complete. It is that during the phase change the energy supplied is used only to seperate the molecules no part of it is used to increase the relating to a result from the motion energy of the molecules so it's temperature will not rise since the relating to a result from the motion energy of molecules remains the same .

I hope this might help u