The tune doesn't do anything wildly out of the ordinary, but it's classic ballad tune nonetheless, and I see nothing wrong with it. Like Anahata, I think you've just got Shirley and Dolly's version lodged in your mind as an ideal, and you need to get away from it. If you're singing it unaccompanied, you've certainly got the opportunity to vary pace and melody, which will give it a new dimension.

I endorse everything Marje and M. Ted say about the need to tell the story. But what story are you telling? Is it the one where the severed head pipes up with a denial of all charges in the last line? I have to say I find this ending a bit trite, and I wonder at what point it was added to the ballad (it doesn't appear to be in Child's versions). But Mrs. Goodyear of Axford sang RVW a version in which the headless corpse "jumped three times upon his legs", so there seems to be an English strand of the ballad involving some kind of Headless Chicken routine.