If the windings above the balls are showing, the bridge may be too thin (was planed) or the underside (the plate) holes are getting over-sized. The balls should prevent this, but if the bridge / plate holes are worn, allowing the string to come up that high, you have a problem. A plate replacement is not cheap, but you can possibly do a little "additive woodworking" to help.
Use a hardwood, like maple, and fashion a popsicle stick of it. Make it just longer than the distance from string-hole one to string-hole six (too long and you hit top braces). Make sure it will fit and cover up the front edge of the chewed-up internal holes, and then glue it in place carefully. The addition of this small amount of wood will not terribly affect the tone, and the new edge will hold the string-balls down.
You could even do two strips*, one for the back-side of the plate holes, too, or even glue the strip ACROSS the string holes and then re-drill (using a drill smaller than the holes, of course -- we're trying to make 'em smaller, after all). If you choose this approach, clamp a piece of junk wood where the drill will come through, or else you'll chew up the wood where the drill point comes out, and end up with a sloppy string hole, which is just what you don't want.
* If you don't think this will work, my old 1964 D-28-with-the-replacement-top has ONLY two such strips (maybe a little thicker, but still...) AS THE ENTIRE BRIDGE PLATE, and it works just fine. People who've heard or played that guitar can attest to that.