It may be due to a worn out or faulty controller. Thumb sticks get a lot of wear and tear and this would explain what you are experiencing. The sprint button works in conjunction with forward movement and if the controller is faulty it would explain that as well. Bungie has also stated they are aware of sprint button input issues that are either delayed or need reapplied for use.
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Also good to know they are working on the sprint issue. I've always had that regardless of how bad the movement issue is and even before I had that issue.
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Nope. Every other game works fine and does it with any controller.