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Doggit
The screws will tighten the handles on both sides, so although the new and old holes may be near each other, it won't weaken the door frame. And as long as you don't make the holes too large, then it should still hold itself side to side. Can you hold the handle alongside the door, and then take a picture. I can see your concerns, but as long as you're careful, then you should be okay.