It may well be the string itself. Sometimes you get wound strings where the core isn't the same thickness all the way along its length or the wrapping isn't quite the same. This affects the mass per unit length, which along with tension and length, determine the pitch of the string. So it may be worth trying a new string. However, it may be that with the heavier gauge of strings used, the non-vibrating part of the string at either end (nut and saddle) is just too long for the amount of adjustment available with the bridge in its current position. There's a small portion of the string at either end which doesn't really vibrate. The heavier the string, the longer this non-vibrating section is, and is why the saddles stagger backwards for the bottom strings.

If you fitted a set of 9s, you'll probably be fine. But you want 11s, (I'm assuming to down-tune a semitone), so I'd maybe try one new low E string and if there are still issues, I can't see any way forward but to shift the saddle back slightly.