The company will turn the matter over to collection and foreclosure if you do not stay current on your payments. Yes, it will affect your credit score, but you can put in an explanatory letter, and it is common that owners cannot sell it (in some instances people have been advertising their time share for sale for $1 for years and been unsuccessful) and creditors may be understanding as to why any sane person might walk away. Alternatively, you may try to gift it to a person who actually is interested in using it or try giving it to a charity as a potential auction gift, etc. In either of those cases, you should make the payments current so the recipient is not assuming a debt, which will cool any ardor in taking it.
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You can stop paying but they can sue you to collect. I suggest you meet with a real estate lawyer and/or a bankruptcy lawyer to see what your options are.
I am a former federal and State prosecutor and now handle criminal defense and personal injury/civil rights cases. Feel free to check out my web site and contact me at (212) 577-9797 or via email at Eric@RothsteinLawNY.com. I was named to the Super Lawyers list as one of the top attorneys in New York for 2012. No more than 5 percent of the lawyers in the state are selected by Super Lawyers. The above answer is for informational purposes only and not meant as legal advice.
Depending on the state, one can "rescind" a timeshare contract but generally it is very close to when you would have signed the contract. Many states have 14 days windows, some less. Echoing others, it seems like you're past the time where you can rescind the contract. The problem is of course is that you're stuck with what you signed. You need to see an attorney right away, and in the meantime find someone to take this timeshare off your hands.
The foregoing is not legal advice nor is it in any manner whatsoever meant to create or impute an attorney/client relationship.