It doesn't matter, because it's a contract. Even if you cancel your card, Adobe can send it to collections, and it will show up as unpaid debt and negatively impact your credit score, which means you might pay more for your next mortgage or car loan.
This is not uncommon for businesses that use annual subscriptions. Certain gyms are particularly known for this. And with Adobe being so sneaky and aggressive about subscriptions, it wouldn't even surprise me.
Damages are generally limited to the extent to which a contract is performed or not performed, and the non-breaching party generally has a duty to mitigate damages.
So, in other words, you can't continue to perform once the other side has stopped performing and then later claim damages for non-performance you were aware of.
It's possible the contract itself specifies otherwise for this situation, but courts are generally not sympathetic to this sort of end-run around common law contract law.
I'm not saying companies have not tried to collect on this basis. I'm saying that if they were taken to court over the practice that they might lose. If Adobe really does this, it might strengthen the case against them.
This is not uncommon for businesses that use annual subscriptions. Certain gyms are particularly known for this. And with Adobe being so sneaky and aggressive about subscriptions, it wouldn't even surprise me.