SusanMomOf2,
Can you twist your whole forearm?  If you are able to twist your forearm so that your hand is palm-up, then you will be able to use the Apache overhand technique.  There need not be a snap of the wrist in that technique.  There also is practically no wind-up -- you just hold your arm at your side, away from the target, and then swing your whole arm up and over, from the shoulder.  The trick is, it is not your shoulder muscles that are providing the force; while you are bringing your arm up and over, you are also twisting your torso around and so the force comes mostly from your torso and abdominal muscles.  The style is not great for distance, but it is good for accuracy.

Of course, at the beginning nobody is any good, so give it some time.  But you already know this, you've probably watched your husband flub a few.  My wife certainly has watched me send a few where I did NOT want them.  Tennis balls into the middle of a blackberry patch, in the middle of her garden, within a foot of the top of my head...

Larry Forsyth

wrote a good description of the technique, along with instructions for a braided sling and some stories about the Mescalero Apache gentleman from whom he learned the sling.