When you select one of the powers, ie. Beast Form, it should de-activate whatever dragon shout you're using. That's what it does for me anyway. Then the same button that activates the dragon shout will activate the power.

You should also be able to favourite the power, just as you do with spells or weapons or dragon shouts, and then select it from the quick-menu (which I think is Y on the Xbox; I think one of the front triggers assigns a thing to the quick-menu when you're in the regular menu).

It could be that I'm completely wrong as concerns the Xbox version, but I can't imagine the functionality is that different, even if the buttons are. Someone here will have the Xbox version though, and will know.