As I dissect your casting problems, I see three major difficulties. The first is a lack of adequate rod loading to cast the heavier fly. The second is a hinging effect, "As I would build the cast the leader looked like a windmill blade with the line-leader connection as the axle." The final problem is the fly hitting you, in effect you describe a type of tailing loop where the fly does not catch the rod leg of the fly line but your head.
A solution to problem #1 is a heavier line. You know that instinctively since you said, "I'm also keeping my eyes open for some heavier gear, 7-8wt range. However, you could use a 6 or even a 7 wt on that 5 wt rod." The reason is that you probably were not casting the full 30 ft of 5 wt line that a 5 wt rod is rated for. A heavier line would both load the rod more effectively and turn a heavier fly over more effectively. So cure #1 is a heavier line when you are making shorter casts.
The cure for the hinging leader is to cut back that tapered leader so that is is short and stiff enough to handle the heavier fly.
The final cure is to get more power into the cast AND to prevent a tailing loop. This is a two step process. One step is to use a water tension cast to load the rod for the forward cast. Let the line and fly go down stream of you and use the pull of the water on the downstream line and fly to add additional load or pull to bend the rod for your first forward cast. It allows you to start the cast with more line out so there is less false casting to extend line. You want to minimize false casting a heavy cone head fly.
The way to avoid a tailing loop and to cast a heavily weighted fly is to use a constant tension cast as suggested above. Search Belgian Wind Cast
for instructions. The constant tension prevents the "rod shock" effect that occurs when casting a heavy fly with alternating forward and backcasts at 180 degrees of separation.