Gonna tap it on this one. Feel free to correct me brychanus and you Andrew
Not sure if you guys have been over it already, but. It seems like you open your shoulders/body too much in the end of the swing (at release) and "spin out". Not by a huge margin, but too much. For me, the hammer is a huge help to remind me of the more "lateral" motion in the throw (English isn't my first language, but trying my best). Posture is a huge deal too.
It seems like you're too stiff and need to freewheel some more. Let the hammer tell your body and arm how to react. When you "catch the momentum" of the hammer in the "pocket/downswing/whatever" and actually feel how you naturally have to brace and the outswing is merely you clinging on to dear life and hoping that you don't launch the hammer 600 feets... That's the BEST feeling ever.. translating that to a disc is another problem, lol.
Disclaimer: you shouldn't just be loose and fling that hammer out, you'll injure yourself. Throwing a somewhere heavy object will normally get your body in a right posture and/or alignment, it just take some self awareness on how your body should react
Gl man!