Our bodies respond more to changes in skin temperature than core temperature. So, if we cool part of the body (for instance with a cold sponge, or cold shower), skin blood flow decreases and skin temperature falls.
Here we “feel” cooler because cold water causes cold temperature receptor activation in the skin. We may also feel more comfortable, as our skin temperature enters the comfort zone. But because there is less blood flowing to the skin, we’ll actually keep more heat inside, thus leading to an unintended overall increase in core temperature.