Still on me, his eyes were blazing with fury.
This sentence is kosher, but I agree it sounds clumsy. I suppose it's because it starts out with a static feeling, perhaps because of the other possible meaning of still (not moving). This contradicts the feeling of movement in blazing with fury -- for example, flames dancing in a fire.
The first sentence, *His eyes, still on me, were blazing with fury," works so much better because the focus remains more clearly on his eyes.
His eyes were blazing with fury, still on me.
As someone explained in a comment, in this case, "still on me" would be describing fury, which doesn't quite work. "Still aimed at me" would work, technically, although in the context this doesn't make a lot of sense.