First things first: consult with an attorney as soon as possible!
Your former employer cannot make you work; but the employment contract you entered into would have to be reviewed thoroughly to determine your rights under the contract, and whether any obligations remain post your employment with said employer. For example, it is not uncommon for employers to include non-compete clauses in contracts, limiting you from working for competitors for a certain period of time after your employment with the particular employer. Also, an employment contract may specify that the employee is responsible for reimbursement of certain costs incurred by the employer if the employee terminates the employment contract in a specified period of time (i.e., moving costs, etc.). But a flat fee penalty sounds a lot like imposition of a mechanism of enforcing involuntary servitude, which is against the law.
Again, you should definitely consult with an attorney before taking any further action.
You should immediately consult an employment attorney who has experience with employment contracts. Check the agreement to see if there is language that says what state's laws govern the contract. Assuming the contract was entered into in South Carolina and is governed by South Carolina law, you should consult with an attorney licensed in SC. I would not ignore the demand letter. Good luck.