First off, the credit union is asking if you are going to pay any of the initial cost of the car without the loan. The money that you hand over for the car from your pocket (not the loan check), is the downpayment. Basically, they want to know if you are going to have any skin in the game.

You do need to keep some cash on hand. I would suggest that you sell one car, use that money for the downpayment, and get the loan check for the rest of the amount. Hand over the downpayment and the loan check to the person you are purchasing the car from, and take the car. Once you get the new car, sell the other car, and keep that cash as a reserve.