If you haven’t already, get several actual offers if you want to know what the car is really worth.
I’d also pull the trade out of the conversation until you’ve agreed on the details of the new vehicle, and at that point you’ll also know the value of your trade.
Out of curiosity did you make them aware of your current balance? Getting a trade offer that matches a payoff triggers the same skepticism for me as payments that end in “99” ($399, $499, etc.)
