When I'm worried about a charge, I use a virtual number generator that my bank offers. It allows me to put a time limit and dollar limit on any charges.
Switching banks frequently can mess with your credit score, so I'd try not to do that too much (especially if you're considering buying a home/car/etc.).
That's not quite accurate. You average account age for credit lines affects your credit score. Banks do not report depository accounts to the credit bureaus because they are not lines of credit, they are liabilities.
They do report to a separate group of risk-scoring companies, but this is mostly to gate fraudsters. Frequently switching checking/savings accounts will have no bearing on your ability to make large purchases on credit. If you happen to close credit cards frequently enough, regardless of whether they are issued by banks, your score could be impacted by reducing your average account age.
Sorry for not being clear — I was just referring to switching banks with whom you have credit cards. I don't know if GP was referring to "banks" in the context of credit or debit, but it seems most people in the US who are savvy about such things use credit cards because of the available rewards.
I think you may find that the swindler may be able to present the old CC number and still charge your account (which hasn't changed). Gym indenture companies are notorious for this.
I had this occur with Capital One and a scam fax-to-email company who refused to accept a trial cancellation. Even after lodging a dispute the scammers could still bill more charges, and I'd have to dispute each one until the bank made a ruling.
Switching banks frequently can mess with your credit score, so I'd try not to do that too much (especially if you're considering buying a home/car/etc.).