If I recently closed my small business LLC and was in a website/marketing contract for 1 year with a company, am I still responsible for the contract?

Eeither way, the company will most likely still be liable for the contract.  If you dissolved the company without first satisfying all of its debts, you may persoanlly liable for doing so. If the company is basically defunct and without assets, stopping payment maybe a good option.  A judgment against a company with no assets is not worth much.  But that assumes that you have maintained the company properly and don't otherwise have persoanl liability for the contract.

If the contract is solely between the website and your LLC you are not personally liable on the debt.  However, if you signed the contract as an individual and/or personally guaranteed the debt then you are on the hook. If this is not clear you should have an attorney review the contract in question.