This is all very simple. Yes, the loan must be paid off in order to sell the car or trade it in. When you trade in your car the dealer is basically agreeing to purchase the car from you and they make you an offer. If you agree they take the money they offered you and give it to the bank in order to pay off your loan. If there is money left over that cash is applied to your new car as additional down payment money, If there is a balance due remaining on the loan that money has to come out of your pocket. In other words if the loan balance on your car is $7500 and the dealer offers you $5000 for your trade in that $2500 difference has to paid out of your pocket.
The key to a successful trade is to determine the current trade in value of the car and then compare that to your loan payoff amount. If the car is worth more than the loan payoff you'll be fine. If the car is worth less than the loan payoff then you have a problem How are you going to pay the balance. If you don't have the cash or can't get the cash, then you aren't trading in your car. at least not now.
Most car buyers who are only one year into their car loan will be upside down onthe loan value (owe more than the car is worth) so trading early is never a good idea.