I have a good idea for you.
Shoot a ray trace out in 4 directions from the centre of your character outwards going forward, backward, left and right as far as the height of the character. Have an animation for the character falling backwards, forwards, right and left. When a character dies, check if there is a ray trace that isn’t hitting anything, if it isn’t, then play the animation associated with the ray trace.
You should make the ray traces respond to world static.
A possible error would be that the characters don’t die realistically, like if you shoot them from the front they will fall sideways instead of backwards, but for a retro game, I would see that as giving it a bit of a retro character 
Just make sure that you have an animation for if all the ray traces ar colliding with something.
You can add more ray traces with more animations and more animations for the same ray traces to make it look a little different every time.