To even consider changing your last name and worrying about the kids' names is an over-reaction.
Although you will feel a "pull" towards the real paternal line, unless there's an absolute NEED to distance yourself from what birth certificates say, then it'd be mad to do so... and, an absolute need would only come from something horrendous uncovered.
The name you have is part of the family quilt ... you've just inspected it a little closer and seen a tiny tear in the corner of one part of that quilt - and, as you look at it, you think "shall I fix that?" ... but then you remember that "granny made that part of the quilt". She made the choice of names at the time; it's your family heritage and inheritance to continue to live with her choice and decision. Note the "should've been" name - follow that line with keen interest ... but leave things as they are because "that's the natural order of things".
My gt-grandma carried the surname of her mother's long-passed husband ... she then married and had my granny. From my granny my mother carried the surname of her mother, not her father as they didn't marry .... it goes on. Who should I be then?