If you don't respect the corporate formalities of your LLC, and you incur personal expenses, or you personally guarantee the debts (which AMEX may require), then you'll be personally liable.
LLCs don't require the same maintenance that corporations do, but they still require that funds don't be commingled, and the the LLC be adequately capitalized, and the various other indicia that mark a true separate corporate existence. Most importantly, if your LLC leaves your LLC's creditors wanting, while transferring assets to you, the sole LLC member, a court would find it inequitable for you to hide behind your LLC shield in not paying these creditors.
Get a "how to" guide explaining how to operate an LLC, and hire a good CPA for help with the finances.
I'm only licensed in CA. Please note that this answer does not constitute legal advice, and should not be relied on, since each state has different laws, each situation is fact specific, and it is impossible to evaluate a legal problem without a comprehensive consultation and review of all the facts and documents at issue. This answer does not create an attorney-client relationship.
The contractual obligation with Amex or any other vendor depends on which name is on the application and agreement. If just the LLC, then most likely only the LLC will be liable for the repayment of the debt. If you fail to observe corporate formalities, then a creditor of the LLC could try and pierce the corporate veil and come after you personally.
If the AMEX is actually issued to the LLC and doesn't require a personal guaranty, you might not have to pay. If there is a technicality with the LLC that allows them to ignore the entity's protection, you might end up being personally liable.