I'm trying to do this now, but the box to pay with a voucher does not show up on their website. I've spoken to Delta online support, but they said I would have to phone in to redeem the voucher. The rep at the Skymiles desk is giving me a hard time trying to redeem this. Still on hold...

EDIT: Success at last. After hours on hold, a lot of contradictory information, and fruitless attempts to point out logical errors, I finally got a representative that was knowledgable enough to apply the voucher. I didn't manage to get the direct ticketing charge waived, though... I was too tired to argue it. Now I just have to figure out what to do with the residual of my voucher, which is about to expire.