Ryan,
Given your description, my vote would be to bring the tree down in a controlled manner rather than it falling down on its own and causing damage. Damaged and diseased
wood that stands nonetheless sounds like a good candidate for the wood pile.
All that being said, from your description (twisted, nearby structures, etc.) perhaps an axe is not the best tool to bring it down. I would suggest first to remove as many branches as possible before cutting the trunk so as to reduce mechanical load on the trunk. Keep the tree as balanced as possible while removing branches/limbs.
Once you have removed as many branches as you can, I would tie a rope/chain as high up as possible on the trunk and then stake that line in the direction you want the tree to fall. Even better, have two lines and stake them so that the trunk falls between them. I would use either a very heavy rope or better yet a nylon tow strap attached to a come along and really tighten those lines before cutting the trunk. At that point you could begin cutting the trunk.
I recently did something like this for a neighbor who had a tree leaning towards power lines. We did use a line, but after we started our horizontal cut, the tree leaned anyways. We pounded a wedge into the cut and reversed the fall of the tree away from the lines. The tree landed exactly where we wanted.
Eric