I have a Mossy 500 that will fail to extract occasionally on 3" Winchester hi speed steel (cheap stuff). It feeds the 2 3/4" stuff just fine and all lead ammo I've shot in it which ain't much because it's a duck gun. The same Winchester ammo jams a friend's 870 Express in either length. She sticks with Kent on waterfowl. So far, my new to me Mossberg 535 works fine with everything I've shot in it, but I've not used it on waterfowl, yet, just dove so the jury is out on the cheap hi speed stuff for it.
Cheap ammo now days can cause extraction problems in some shotguns regardless of brand. I've heard of folks having the same extraction problems with Winchester's price point dove loads, but i've not seen that do far. No guarantee that the Mossy will function any better with whatever ammo you're shooting. I'd try a variety of ammo in the Winchester first. Ammo could be the problem, not so much the gun. Not sure what your gun's doing, of course. It could also be something a gunsmith could fix. The 1300 is a quality pump IMHO. I wouldn't just up and give up on it until I'd exhausted all other options.