offered to pay a credit card in full

I don't think it sounds true. Creditors often hire a collection agency when they are not able to recover the balance. This way, they don't have to put too much time on your file. Maybe, you will hear from some CA that happens to be the worst in the industry. Double check the laws with your state attorney general's office.

I think you would need written proof of their refusal to accept your payment, or something other than just your word against theirs in a phone conversation.

If you can get something like that in a recorded conversation, you might have a leg to stand on if it went to court. However, you need to know the laws regarding recording phone calls in your state. Doing so without following the law can get you into deeper trouble than you have now.