That very “Wells Fargo fraud department” scheme just happened to my wife a few weeks ago. Luckily she was working from home and I overheard the convo. I ran into the room and disconnected her call. We closed our accounts out after painfully dealing with the real WF fraud department, whose Indian accent was punishingly thick and their script was very unhelpful. We had to keep repeating that we wanted our accounts locked and to disable wire transfer capability. It was surreal how bad it was.
The Sharks ARE out there. Even after my first very costly experience, I found it somewhat difficult to realize in the internet age, such vicious, greedy people are never very far away, and one has to always be on your guard when it comes to talking about your savings - with anyone.
Keep the small talk short, if at all. Never offer any ‘extra’ information. Very few banking decisions need to be done “Right Now!!”. Always give yourself time to think things over or to verify any change with another branch of that bank. I’m not talking about Brokerage accounts.