Just give it a try yourself. Place an ad on tug2.net or someplace else. A word of warning though, use a separate e-mail address because you may get inquiries months after you have sold.
I bought my first contract privately and the seller said he was bombarded by interested buyers for a long time even after our transaction was completed. This was over 5 years ago.
Once you find someone interested, just contact one of the timeshare closing companies to take it from there. Some even provide a sample sales agreement you can modify. Dean has mentioned that it can be done without a closing company. But since the closing is usually paid by the buyer and the closing company handles paperwork that you would instead need to do, it would be easier for a novice to use one.
If you can not find a serious buyer on your own through the internet ads, you can always use a broker then. A broker puts the buyer together
with the seller. After that, the paperwork is handled by the closing company.
Good luck.