Also before you mess with the limit screws,
check your cable tension. You are having symptoms of slow shifting and too much chain rub. That could be a limit issue, but I'd be hesitant to go there first given the bike's propensity to throw the chain off the outside. Try testing cable tension first.
Hand shift by pulling the cable. Does the rubbing stop when you are pulling on it? If so, then maybe you just don't have the cable tight enough such that when you've got the lever pulled all the way, you aren't getting full movement of the cage (but you can get the additional necessary tension by pulling with your hand).