Having problems with truck starting on 2005 Lincoln Navigator

Why do the power stays on when my 2005 lincoln navigator is turned off? The radio is turn off but the light to the cd player is still on and you are able to push the radio button and it comes on even when the car if off. This sometimes allow the truck not to start in which it clicks when you try to start it. I have a new battery which was replaced 2 weeks ago. Would this cause the battery to drain and the truck not to start? or would there be something wrong with the connectors?