The credit card company "could" come after you personally, not just the LLC, for the money owed on the business credit card. Many attorneys have LLCs as well and the credit card companies make sure that we sign off individually as personal guarantors for the credit card as well.
The information provided in this answer does not create an attorney-client relationship. If you are interested in his legal services, feel free to call Chris at (303) 409-7635 at his law office in the Denver Tech Center. All initial consultations are free of charge.
This is simple -- the credit card company wants their money and they will come after you personally if you are a signer in the account. Why don't you save a headache and negotiate a resolution?
In no way am I offering you legal advice, and in no way has my comment created an attorney-client relationship. You are not to rely upon my note above in any way, but insted need to sit down with counsel and share all relevant facts before receiving fully-informed legal advice. If you want to be completely sure of your rights, you must sit down with an experienced criminal defense attorney to be fully aware of your rights.