I mean, if you truly need a car and this car fits your needs and is in good shape, you might consider just keeping it at least until the car market cools off. Used cars are pretty highly priced right now. You've already taken the biggest depreciation hit by holding it for the first year, holding it for a couple more years isn't going to depreciate as much. I understand you don't want the payment but depending on your car needs over the next 5-10 years, it might not be a bad idea to just keep it, enjoy the fact that it is under warranty and should be very reliable for at least the next 5 years. The sunk cost is already realized, don't beat yourself up or rush to sell it thinking it will undo the frantic decision you were forced into last summer.
If you decide to sell, I like to get an estimate from carmax first. They are the easiest route but that simplicity comes at a cost of getting a potentially lower offer than you could get selling it privately. You don't need to pay off the loan if you're selling to carmax or another dealership. I've never personally done a private party sale with an outstanding loan so I can't speak to how that works. As far as payments, carmax or a dealership would write you a check. I would be a little more cautious selling it to a stranger on facebook in terms of payment.
