Definitely don't let them run their script and mess you around.
Personally, I tend to start any call with a customer support person with something like "Hello, I'm calling about an existing account, would you like the account number?". Almost invariably, they will say yes and then do a couple of checks to confirm my identity for data protection purposes.
At this point they know exactly who I am, and they probably ask why I'm calling. If I'm asking to change or cancel something, I state that immediately, so now they have a record of both my identity and my request.
By now, I'll also have a pretty good idea of whether I'm talking to a human being who is interested in helping me or some call centre automaton who's just going to be awkward and stick to a script.
If they then try to give me the run around, I just repeat that I've informed them of my details and my wish to change/cancel, if necessary state that I'll be cancelling payment authority accordingly, and then tell them I'm ending the call.
If possible, I'll also keep a recording of the call (which is legal in my country) but at the very least I'll make an immediate written record of the time of the call, who I spoke to, and what was said.