If I arrange to donate jewelry appraised (documented) at $10K to a 501C3, and receive cash from them for 50% of it's value, can I simply claim a jewelry donation of $5K?

from IRS 526 526 

donation of capital gain property

Amount of deduction—General rule. When
figuring your deduction for a contribution of cap
ital gain property, you can generally use the
FMV of the property.
Exceptions. However, in certain situations, you must reduce the FMV by any amount that would have been long-term capital gain if you had sold the property for its FMV. Generally, this means reducing the FMV to the property's cost or other basis. You must do this if:
1. The property (other than qualified appreciated stock) is contributed to certain private nonoperating foundations,
2. You choose the 50% limit instead of the 30% limit for capital gain property given to 50% limit organizations, discussed later,
3. Not applicable
4. Not applicable
5. The contributed property is tangible per sonal property (defined earlier) that:
a. Is put to an unrelated use (defined later) by the charity, or b. Has a claimed value of more than $5,000 and is sold, traded, or other wise disposed of by the qualified or ganization during the year in which you made the contribution, and the qualified organization hasn't made the required certification of exempt use (such as on Form 8282, Donee Infor mation Return, Part IV). See also Re capture if no exempt use, later.

in the worse case, your deduction would be 50% of the tax basis and you would be treated as having sold the other 1/2 for the $5K you get

even in the best case getting cash back could be treated as selling 1/2 of whatever you give