I've only felt something like that when I was badly dehydrated and then the heat wore me down. Having the cold drinks would benefit if it encourages you to drink enough, but I would be leery of depending on it to help cool your core temperature. It doesn't take that much of the heat to raise the temperature of that amount of water, so it physically cannot cool you that much.
I think that the thermosensors in the throat and mouth interpret ice-cold water as a cool temperature and inhibit the sweating as a response. That may or may not be what you want to happen.