Can I trade in a car I still owe money on?

Yes. The dealer pays off your existing loan as part of the deal. If the trade is worth more than the payoff, the difference is equity that reduces your new purchase price. If you owe more than it's worth, that negative equity is usually added to your new loan, increasing the balance and payment. Get a written payoff quote from your lender and confirm the dealer's payoff on the contract.