I assume OP is in the US. In the UK at least there are companies that will purchase cars from individuals. The companies either sell them on to dealers, or make repairs as needed and sell themselves. It's a very quick way to sell a car (can book an appointment for next day etc.) and the companies generally guarantee to give you a price, and unless you outright lie to them they know that they can't come complaining to you afterwards if more issues are found (unlike a private buyer might want/try to). Downside is that you might be able to sell the car for more privately and/or these companies tend to lower offers significantly for any damage/repairs needed. So if your car really has to go quickly and there are similar services in the US, that could be a way to go. I'm not clear about the loan you have, is it just a generic loan that you have used the money to buy the car (so you can sell the car and continue to pay the loan), or is the loan specifically against the car (loan provider owns the car until you pay off loan, so you will have to pay it off before selling the car?).
I used one such company to sell my car earlier this year, process was very smooth, took 3 days end to end, and price was reasonable. As that was the first time I have ever sold a car, will be years before I do so again (currently don't have a car, so that's if I do so again), the effort to sell a car privately and potentially get a bit more didn't seem worth it.