>I learned this the hard way once when I tried to get rid of a $5 subscription I didn't understand with Discover. I naively figured I would just request a new card and that would be that

That's a bad idea anyways. Just because your credit card was declined doesn't mean you're off the hook. Most businesses are reasonable and won't chase you further, but some persistent ones (eg. gyms) are known to continue your subscription and send your bill to collections.