All dealers will pay off your trade. Its just a matter of how its done. the ads likely say "We will pay off yoru trade when we make a deal" or somethign to that effect. Well of course they will pay off the trade if you make a deal, otherwise you would not have a deal.
Its the same from Maine to Spain. When you trade a car in, they will put a value on the car. If you owe more than its worth, then you need to pay the shortfall either in cash, or by rolling that shortfall onto the next loan. If you owe less than its worth, the excess is applied to the new loan as a down payment. IN either case, the dealer is paying off the loan for you, otherwise they could not take the trade.
Dealers often advertise with phrases such as:
"we will pay off your trade no matter how much you owe if we make a deal"
"Push pull or drag it in, any car is worth $5000 this weekend"
my favorite? "No credit applications will be refused" - they can't refuse your APPLICATION. They can, however, decline to get you financed.
The devil is in the details