> Is the bank going to call around to services that I have paid for previously and tell them my new credit card details? I hope not

Yes. Anyplace that bills you regularly on the old number will automatically get the new number.

> You can't continue to use the old number for new transactions.

It depends on how you define new. A new biller, then correct. But if it's a biller that you have an existing relationship with, i.e. they bill you every month, then they do send them the new numbers (and/or allow them to use the old number).

I can't say if this is a global policy of all issuers, but I can say that I've experienced it. I've also seen it in the fine print when I signed up for repeated billing.