Dave Ramsey answers this question every other day on his show. His advice is always to sell the car privately and pay the difference, as painful as it might be.
Dave's advice tends to be "rip the bandaid off quickly" advice. It doesn't always make perfect sense mathematically. With a leased car, the financial implications are such a shitshow I'd be inclined to get the whole problem out of my life ASAP, take the fastest option as a form of harm reduction, and go with the plan of selling the car privately and making up the difference, especially if I had the cash to do so (if not, he always recommends trying to get a loan for the $6k shortfall through a credit union, if possible).
Selling a car that is worth more than $10k through private sale is a little tricky. I'd recommend ebay and Craigslist first--take awesome pictures and advertise that thing well. Price it competitively. Read up on avoiding scams. Any kind of professional car selling operation is not going to offer you the best deal (they can't; they have to make money when they sell the car), and you need the money so you have to at least try to deal with the hassle of selling it yourself.
You're going to take a haircut on this deal. But hey, you'll never lease a car again.