State Taxes for Raffle (Gambling) Winnings

I recently won the 50/50 raffle at a sporting event in Illinois but I live in DC. I want to make sure I put aside enough to cover the state taxes that will be due in 2020. The check for the winnings already had 24% deducted presumably for the federal taxes estimate. 

From my research, I gather that I will still have to report the winnings as income but the Fed taxes that they already took out should cover the fed portion (or is supposed to pending my actual tax bracket). My question is, will I have to pay both the IL gambling tax (5%) in addition to the DC gambling tax (8.5%) or will it by 5% to IL and then it is credited to DC and only have  3.5% as additional?

I think I will still have to file the W2G with IL but DC will tax the whole thing as income so i am wondering if they will credit what is already paid to IL. Basically, should I set aside 8.5% or 13.5% of the total winnings to pay for state taxes during 2020 tax return time. 

Thanks!