Don't trust cancelling your card either. I closed my account at Capital One, paid the final balance, and six months later I noticed a steep drop in my credit score. I had a $3 monthly charge that kept recurring even though I had closed my account.
Also, because my account was "closed," I didn't receive any statements notifying me that I was being charged. I only discovered this issue when my credit score dropped by 100 points.
Closing a personal credit card, in my experience, temporarily drops the score a few points and then it goes back to normal. It's a myth promulgated by banks to keep accounts open.
If so, that just raises the question: of what benefit is it to the banks to keep unused accounts open? The maintenance costs may be low, but they're still nonzero.
Also, because my account was "closed," I didn't receive any statements notifying me that I was being charged. I only discovered this issue when my credit score dropped by 100 points.