I won an existing gift

If you ask within the giveaway, the GA creator can just link the comment to staff, and staff will be able to approve the reroll quickly.
Just keep in mind that staff is slow in dealing with requests these days, so it may take a while to process.
It's considered polite to leave the giveaway unmarked on feedback, unless the giveaway creator doesn't give you a response to indicate they're following up on the reroll request, at which point you should mark Not Received once you're able.

Note that you can hide giveaways you own by clicking the "eye" symbol next to a giveaway's name in the giveaway list, or in the center-top of the page when looking at a giveaway directly. If you can't find a giveaway up for the game you want to hide, you can do so by clicking the Archive button in the bottom of the page and searching out the giveaway there.
You can close a thread once you've gotten all the answers you want out of it by clicking the green "Open" in the top right corner.

Hope that helps!