Yes and No, The statement says $5,000 because that is what you won and what needs to be reported to the IRS. So you will pay taxes on the full $5,000 to the Federal just like you did to the state.
You are not being taxed twice by the state as you are receiving a credit for that $200 paid towards what you would owe the state. However, if the $200 was not enough withheld when added to the rest of your income you may need to pay any additional amounts due. Just be sure to enter the $200 paid when entering your W-2G.
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