Can I contact a credit card company and try to negotiate a settlement for an old card balance

Since the debt is already charged off, there is really nothing that you can do about it. It is going to be on your credit reports and it will hurt you for some time to come. The best way to get your credit score up now, is to slowly rebuild good credit, maintain a low balance on your current credit cards, and don't be late on any payments.

If you haven't paid them or promised them any money towards the delinquency, and they have not contacted you, then you should just leave it alone. You didn't say what state you're living in, but every state has a statute of limitations on how long outstanding debts can be collected. Usually the statute of limitations is anywhere from 5 years to 9 years. If you simply wait it out past your state's statute of limitations, you will no longer have any obligation to pay off the debt, even if they come after you for it years later.

Remember though, that if you send them payment or promise to send them payment, you restart the clock on the statute of limitations.