If your water has a rotten egg smell, you should have your water tested for sulfer. I do not know where you are located but, if you live in an area where the water table is low, you may be drawing water from a different vein that has a high content of sulfer.
I believe sulfer can be removed by a magnesium filter. You should check.
You should be able to detect it in the cold water as well but not as much because as you heat the water, it activates the sulfer and also the smell is carried around by the steam.