We also spent a bunch of time on the phone with TWC directly, and they had no account in our name or any history of our account.
When it was time to cancel I learnt that I was actually on a yearly contract - this was a month later so past the cancellation period. The support advisor claimed that I not only received an email but actually opened it, at which point I knew they sent it to the wrong address or were outright lying as my email client never loads remote resources, making it impossible for them to see that I "opened" it. He also refused an IMO reasonable demand of reviewing the call recording (it was just a month ago so they should still have it) to determine who was in the right.
In the end, I simply blocked further payments. When collections reached out, I explained the story above and they went away immediately. Collections ended up having significantly better user experience than the original company.
Best part of this story
Theses systems were built by the creditors not the consumers.