The light on the charger flashes when it first connects, then will be red when charging and green when it's charged.
The light on the EGO should turn on every time you press it, the 5-click on/off feature is just for when you want to put in a pocket and make sure the button isn't pressed accidentally.
It sounds like you're having a different problem if the light is coming on when you press it. If you hold the stardust beside your ear and press the button for a second, can you hear a bit of a sizzle ?