That is not a fun problem to have. I'll try to help if I can from my experiences. I'm guessing your horse probably has a forward girth groove (right behind the elbow), flat shoulders and low withers. Either that or a flat-sided straight body with low withers (like a mule).
I had to try many, many saddles to find one that worked for my mare.
The non-slip pads have not worked well for me because I have never found a saddle that did not move *at all* and so the non-slip rubs hair and hide off as they move. I have also tried a crupper and I did not like it. It did not rub much under her tail, but my horse did not care for it.
What has helped me the most was to find a saddle that had the cinch set so it hangs down right where her girth groove is. In other words, a lot of saddles have the cinch farther back where most horses have their girth groove i.e. 3-4 inches behind the elbow. So you end up cinching the saddle farther back because if you put the cinch in the appropriate place for your horse, the front of the saddle would be on the horse's withers and shoulder. Cinching farther back means it is on a wide part of your horse's barrel and the cinch will slide forward as the horse moves, especially down hills.
I also found a saddle that fitted her back as exactly as I could, especially matching the angle of her back behind the withers. Be sure your saddle is wide enough. Y-rigging also helps on either an English or Western saddle.
If you ride English, put the girth on the front two billets instead of the back two.
Hope something here is useful to you.