You aren't being double dipped. You're getting $10,000 worth of watches. You can keep them or sell them, but either way you have to pay taxes on them. *However*, I'm don't think the foundation can make you pay the taxes up front. If you won cash, they might be required to withhold some for taxes. Since there's no way to withhold 25% of a watch, they have no duty to send any money to the IRS. Since they don't have a duty to send money to the IRS, they can't make you pay the income taxes on the watches to them before giving you the watches.
They will be sending you a 1099 for $10,000 that you have to include on your tax return. I strongly suggest you pay a couple jewelers for appraisals on the watches. Just because the alleged "retail value" was $10k doesn't mean you would have had to pay $10k for the watches. Jewelry is always "on sale", meaning it's artificially marked up then discounted to what people will actually pay. If $10k is the marked up price, the actual retail value could be $7,000. You could also argue that their value is whatever you can sell them for. I recently attended a charity auction for Children's Hospital Foundation in Seattle & saw jewelry with a "retail value" of $5,000 selling for less than $2,000. And that's from people who wanted to overbid because it was for charity. If you report less than the 1099 amount, you'll have to include copies of the appraisals to explain the variance.
You didn't mention the name of the charity, but I wonder about a charity that wants you to give them money for taxes. How do you know they'll send that money to the IRS? Are you sure they are a legitimate 501(c)(3) entity? The scenario you describe is sending off warning bells in my head. Either this 'charity' is a scam, or they don't know what the heck they're doing. Either way, do not send them money.
This post does not constitute legal advice, nor does it create an attorney-client relationship. Postings are based only on the information provided and you should consult an attorney in your area before relying on information contained in this post.