I had the same problem. I'm tall with a large frame, and I just wasn't tying it tightly enough in back. It needs to be a seriously tight double knot so it won't gradually slip loose.
You can't imagine how frustrated I was the first time I ventured out for a long walk with my 3-month-old in the Moby and my 4-year-old tagging along. We were stuck walking along with a sagging, crying baby and it was no fun at all! I really learned about tying it tighter the hard way - LOL! I like my Moby now, especially when we're just hanging out in the back yard, but I actually prefer my ring slings. So much more versatile, and great for nursing, which DD wants to do constantly!
Good luck with your Moby. I think they're really cool.
... and when they say tie it tighter, that means tighter than you think it could possibly need to be. At each step (when you go over your shoulders, tuck, etc.) just pull pull pull and give a little bounce if that helps you get it tighter. Don't worry, your baby will still fit when you stretch the wrap to put him in.
Otherwise, if it doesn't work for you, you might need a non-stretchy (woven or gauze-y) wrap for more support!
Tying TIGHT and spreading fabric FLAT AS POSSIBLE are the keys (when fabric is all bunched up, it's never as tight as you think). I can still carry an over 20 pound baby, an eight month old, for hours in my Moby without stretching problems. So I see no reason why a three month old is going to be a problem.