I've never used an ice ax. Do you normally use two hands to anchor it into the snow/ice?
There are many correct ways to hold an axe, depending on the terrain. On moderate terrain, you usually hold it like a cane, with the spike (the end of the handle) sticking into the snow. You can hold it by the head in the “low dagger” position, sticking the pick into the snow at waist level. You can hold it by the head in the “high dagger” position, sticking the pick into the snow above your head. On technical ice, you hold it by the handle and swing the pick into the ice. For self-arrest, you use two hands and stick the pick into the snow to stop. You also use two hands to control your speed in a glissade. It’s best to get some instruction before going out with an axe.